THEORY OF ERRORS AND LEAST SQUARES THE MACMILLAN COMPANY NEW YORK • BOSTON • CHICAGO • DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO., Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO THEORY OF ERRORS AND LEAST SQUARES A TEXTBOOK FOR COLLEGE STUDENTS AND RESEARCH WORKERS BY LeROY D. WELD, M.S. PROFESSOR OF PHYSICS IN COE COLLEGE Neto gork THE MACMILLAN COMPANY 1922 All rights reserved 9 7sr IV-/^ S^ A ^ ^^^..:^ COPYKIGHT, 1916, By the MACMILLAN COMPANY. Set up and electrotyped. Published March, 1916. Eclucair^on OtpT- NorbjooD iPress J. 8. CushinK Co. - lieiwick & Smith Co. Norwood, Mass., U.S.A. PREFACE There are few branches of mathematics which have wider applicability to general scientific work than the Theory of Errors, and few mathematical implements which are capable of greater usefulness to the research worker than the Method of Least Squares. Yet, for some reason, students are rarely given opportunity to acquire facility in these lines, the result being that too many of our scientists and engineers go about their work without such equipment. It would be almost impossible to enumerate the variety of ways in which the ideas relating to these subjects adapt themselves to even such simple bits of quantitative work as the chemist or the surveyor is daily called upon to do. And it is difficult for the writer to imagine how an elaborate research in any of the exact sciences can be carried on at all, without the constant application of these principles throughout both the preliminary and the final stages of the work. The satisfaction to be gained from the application of the theory of pre- cision alone is well worth all the time necessary to acquire these subjects. Add to this the fact that the 5004'. a VI PREFACE theory of error distribution has direct theoretical bear- ing upon certain very important laws and problems of physics, chemistry, astronomy, and even of biology, and the reasons for students' having opportunity to attain the elements of the subject become still more emphatic. This small volume embodies the material used by the writer as lecture notes during the past twelve years. It is intended as a presentation of the Theory of Errors and Least Squares in such a simple and con- cise form as to be useful, not only as a textbook for undergraduates, but as a handy reference which any research worker can read through in an evening or so and then put into immediate practice. It will be noticed that the illustrative examples and problems are drawn from various branches of science, suggesting the wide range of possible application. No attempt is made, of course, at an exhaustive treatment in such small compass. Some of the special methods employed by expert computers, often included in larger works, have been purposely omitted. For the conven- ience of the student, and in order not to interrupt the thread of the subject, a few of the more complicated mathematical discussions have been set apart in the Appendix and referred to at the appropriate places. It is not intended that they shall be omitted from the course when using the book as a text, though the cas- ual reader may get along ver}^ well without them. The writer wishes to express his appreciation to the numerous friends who have kindly given aid by way PREFACE VU of furnishing data for the illustrative examples, or otherwise. Where material has been taken from other works, due credit has been given for the same. L. D. W. Cedar Rapids, Iowa, December, 1915. CONTENTS CHAPTER I ON MEASUREMENT ARTICLE PAOB 1. Definition of measurement 1 2. Indirect measurement 1 3. Estimation 3 4. The impossibility of exact measurements ... 5 5. Errors of measurement 6 6. Exercises . . 8 CHAPTER II ON THE OCCURRENCE AND GENERAL PROPERTIES OF ERRORS 7. Errors and residuals • ,11 8. Classification of errors . 13 9. Mistakes ,17 10. General methods of eliminating persistent errors . . 18 11. Exercises leading to an understanding of error distri- bution 22 12. Remarks on the distribution of errors .... 26 13. Precision 28 14. Mathematical expression of the law of error . . . 30 CHAPTER III ON PROBABILITIES 15. Fundamental principle . . ... . . .31 16. Definition of mathematical probability .... 32 17. Permutations .34 X CONTENTS ARTICLE PAGE 18. Combinations 36 19. Probability of either of two or more events ... 39 20. Probability of the concurrence of independent events . 40 21. The coin problem 41 22. Important exercise 44 23. Empirical or statistical probability 45 24. Exercises 45 CHAPTER IV THE ERROR EQUATION AND THE PRINCIPLE OF LEAST SQUARES 25. Analogy of error distribution to coin problem 26. The most probable value from a series of direct measure- ments. The arithmetical mean 27. Gauss's deduction of the error equation . 28. Discussion of the error equation 29. The principle of least squares in its simplest form 30. Exercises 51 52 56 58 61 CHAPTER V ON THE ADJUSTMENT OF INDIRECT OBSERVATIONS 31. Observations on functions of a single quantity . . 65 32. Observation equations for more than one unknown quantity 67 33. More observations than quantities. Normal equations . 69 34. Reduction of observation equations of the first degree . 72 35. Illustrations from physics 74 36. Illustrations from chemistry 78 37. Illustrations from surveying 81 38. Illustrations from astronomy 85 39. Observation equations not of first degree ... 89 40. Observations upon quantities subject to rigorous con- ditions 91 41. Exercises 94 CONTENTS XI CHAPTER VI EMPIRICAL FORMULAS 42. Classification of formulas 43. Uses and limitations of empirical formulas 44. Illustrations of empirical formulas . 45. Choice of mathematical expression . 46. Exercises PAGE 104 105 107 111 115 CHAPTER VII WEIGHTED OBSERVATIONS 47. Relative reliability of observations. Weights . . 124 48. Adjustment of observations of unequal weight . . 126 49. Exercises 128 50. Wild or doubtful observations 135 51. The precision index h 136 52. General statement of the principle of least squares . . 139 CHAPTER VIII PRECISION AND THE PROBABLE ERROR 53. Discontinuity of the error variable 54. Value of the integral /, and the relation between c and h 55. Probability of an error lying between given limits. The probability integral 56. Calculation of the precision index from the residuals 57. Approximate formulas for the precision index 58. The probable error of an observation 59. Relation between probable error and weight . 60. Exercises 61. Probable errors of functions of observed quantities 62. Probable errors of adjusted values .... 63. Probable errors of conditioned observations . 64. Exercises 141 143 144 146 149 152 155 159 163 166 170 171 Xll CONTENTS APPENDIX SUPPLEMENTARY NOTES PAGE A. Proof of the necessary functional relation assumed in deriving the error law. (Supplementary to Art. 27) 177 B. Approximation method for observation equations not of the first degree. (Supplementary to Art. 39) . . 178 /.OO C. Evaluation of the integral | e-^^^dx. (Supplementary to Art. 54) 180 Z). Evaluation of the probability integral. (Supplementary to Art. 55) 182 E. Outline of another method for probable errors of adjusted values. (Supplementary to Art. 62) . . . 183 F. Collection of important definitions, theorems, rules, and formulas for convenient reference .... 185 THEORY OF ERRORS AND LEAST SQUARES THEORY OF ERRORS AND LEAST SQUARES CHAPTER I ON MEASUREMENT 1. Definition of Measurement. — To measure a quan- tity is to determine by any means, direct or indirect, its ratio to the unit employed in expressing the value of that quantity. Thus, in measuring a Hue, we find that it is a certain number of times as long as the foot or the centi- meter, and this number is said to be its value in feet or centimeters. This definition must be clearly understood to be in- dependent of whatever process is used in the measurement. We could measure the area of a polygonal piece of sheet iron in two ways : either by measuring its sides and angles and computing its area by geometry, or by weighing it and comparing its weight to that of a square piece with unit side. Either of these processes is a true measure- ment of the area, though neither is a direct measurement. 2. Indirect Measurement. — Indeed, with the excep- tion of one kind of magnitude, very few measurements are direct. By this is meant that we do not, in general, B 1 ;2 THEORY OF ERRORS AND LEAST SQUARES apply the unit of measure directly to the magnitude to be measured. This is done commonly only in the case of length. We can, in measuring a line, apply the yardstick directly along the line and determine in this way how many times greater one is than the other. But we cannot take a lamp in one hand and a standard candle in the other and determine the candle-power of the lamp in any such direct manner. So far is the above mentioned principle true, that, as a matter of fact, nearly every kind of measurement is made to depend, in practice, upon measurements of length. This will be clear from a number of illustrations. Angles are measured, not by applying the wedge-like degree as a unit, but by measuring the length of the arc laid off on a curved linear scale, or by measuring the lengths of straight lines connected with the angle and com- puting the latter from its trigonometric functions. Time is measured, not by counting the minutes and seconds in the interval, but by observing the motion of the clock hands over a curvilinear scale called the dial, marked off in spaces of equal length; or by noting the lengths marked off on the chronograph record by a pen point which is given a lateral jerk electrically at the beginning and end of the interval. Every magnitude measured off on a dial is finally referred to length, as exemplified by pressure gauges, gas meters, electric me- ters,, aneroid barometers, etc. Temperature is measured off as a hngth on the stem of the thermometer. ON MEASUREMENT 3 Atmospheric pressure is measured, and even expressed, in inches or centimeters of mercury. Weight is measured, in the final adjustment, by the position of a sHde or rider on a linear scale, or in refined work by the position of the balance pointer at equilibrium, the sensibility of the balance being known. The common spring balance and its more refined near relative, the Jolly balance, illustrate the linear principle in another way. In short, every measuring instrument has some sort of linear scale, either straight or curved, on which some sort of indicator or pointer moves. The reason for thus referring every kind of measurement to a simple one of length is mainly the one already referred to, that length is the only kind of magnitude that can be conveniently compared directly with its own unit. But there is another reason. The eye can estimate a length with far greater accuracy than the muscles can estimate a weight, the hand a temperature, or the con- sciousness an interval of time ; and this process of esti- mation plays an all-important part, as will now be seen, in every kind of accurate measurement. 3. Estimation. — The degree of precision with which an observer can read a given linear scale depends upon two things, namely: (1) the definiteness or sharpness of the marks on the scale and of the pointer or indicator, and (2) the skill with which the observer can estimate fractional parts of one interval or scale-division. The former item may be made clear by comparing the 4 THEORY OF ERRORS AND LEAST SQUARES scale and indicator of an ordinary spring balance with those on a delicate ampere meter or aneroid barometer; or the graduations on a surveyor's leveling-rod with those on a silver-inlaid standard meter bar. As to the second matter, it is of the utmost importance that the observer drill himself in this process of estimation. In no case does the accuracy of a single scale-reading end with the fineness of graduation of the scale, providing the scale lines and indicator are sharp and distinct; it can always be carried a step farther. It is the custom of practical observers to make estima- tions of fractional units in tenths, not in halves, thirds, etc., and to record the readings decimally. No attempt is made to estimate the hundredths, unless it appears to the observer that the fraction is exactly one-fourth or three-fourths, when he would be likely to record .25 or .75 ; even this is a doubtful practice. The reading of any linear scale may be carried, in general, to an accuracy of one-tenth of the smallest scale-division by the estimation of the eye alone, or the eye aided by a magnifier if desir- able. In many instruments of precision, the linear scale is provided with some sort of vernier, which is a mechanical substitute for the estimation of fractional parts of scale divisions. Descriptions of the different kinds of verniers in use may be found in any elementary laboratory manual of physics, or in any encyclopedia. But even the use of the vernier requires the same sort of skill and judgment as estimation, namely, a correct idea of linear position and ON MEASUREMENT 5 coincidence. And in the vast majority of measuring instruments, no vernier is provided, and the observer must be able to estimate tenths accurately and without hesitation. 4. The Impossibility of Exact Measurements. — Every scientist is familiar with the fact that there is no such thing as an absolutely exact measurement, for the simple reason that the quantity measured and the unit of measure are never commensurable. If we weigh carefully a small piece of metal on a common balance, a typical result would be 3.9843 grams, and not a whole number, as four grams. This is, however, only an approximation to the true weight, even if correct to four decimal places, just as the number 3.1416 is only an apn proximation to the value of tt. If a more sensitive bal- ance is used, the result may be 3.984326 grams ; but as the masses of the piece of metal and the gram weight are in- commensurable, the true weight, even if it were possible to weigh without the inaccuracies that arise from im- perfect apparatus and judgment in estimation, would be inexpressible in grams, and the result obtained could be true only to the degree of approximation represented by six places of decimals, that is, to the nearest millionth of a gram. * What is true of weighing is true of all measurement, and it will readily be seen that to obtain the true value of any actual concrete quantity is as hopeless as to obtain the true value of V2^ or tt, or logio 17. 6 THEORY OF ERRORS AND LEAST SQUARES 5. Errors of Measurement. — Aside from the mere incommensurability of magnitudes, there is another and far more serious hindrance to the obtaining of correct values by measurement, and this is what is technically known as error. Suppose the bit of metal, which was found on the more sensitive balance to weigh 3.984326 grams, be now weighed again, by the same person, in the same room, on the same balance and with the same weights. More likely than not, the result will turn out to be different from the former result by some millionths of a gram, perhaps thirty or forty millionths. This means simply that neither re- sult is correct, even to the sixth decimal place. Again, if we go out with a surveyor's transit of the finest construction and measure with the utmost care, to seconds even, each of the three angles of a triangle marked out by accurately centered stakes on level ground, and add the three results together, we shall probably find that their sum differs from 180° by several seconds one way or the other. We may repeat the operation with equal care and skill, and get a still different result, perhaps farther from 180° than the first. This illustration will be all the more striking, in that in this case the true value of the sum of the three angles is known from geometry, while in the case of the weights the true value is not and can never be known. Even here, the individual angles cannot be obtained exactly. The causes of error in precise measurements are many and various. A single example will suffice to illustrate ON MEASUREMENT 7 this. Suppose we wish to measure the distance from one stake to another with a surveyor's chain. Two men carry the chain. Each time they advance, one adjusts the following end to the rear marking-pin, the other sets a new pin at the leading end, and neither can do this work with absolute accuracy. They do not stretch the chain tight enough; they do not hold the chain horizontal in going up or down hill ; they do not follow a straight line ; they do not notice kinks in the chain, and they neglect the fact that the chain is wearing at the joints and getting longer. As a consequence of all these small items, and many others not mentioned, the measurement may in the end be several inches from the truth if the line to be measured be very long. This is only one instance showing how hundreds of little disturbances may combine and form one final resultant error which may be positive or negative, great or small, according to which kind of dis- turbances predominates (that is, whether they tend to make the result too large or too small), and to whether they happen to be about evenly balanced or not. A systematic study of the occurrence of errors gives rise to a mathematical analysis, based essentially upon the principles of probability and known as the Theory of Errors; and our attempts to apply this theory to the re- sults of measurements, with a view to getting the values that are probably nearest the truth, have resulted in the formulation of certain rules embraced in that part of the error theory known as the Method of Least Squares. 8 THEORY OF ERRORS AND LEAST SQUARES EXERCISES 6. The following exercises are intended for the use of students who have not done much laboratory work nor had the advantage of a course in laboratory measurements or field work. It will be seen that they are largely sug- gestive, and they may be modified as desired to suit the circumstances. For advanced students and research workers they may be omitted altogether. 1. Can you think of any kind of accurate measure- ment not ultimately employing some sort of linear scale ? 2. Show wherein the following kinds of measurements are made to employ a linear scale : area of a piece of land ; density of a solid ; relative humidity of the atmosphere ; index of refraction of a transparent substance ; volume of liquid from a burette. 3. Determine the volume of a material sphere, cylinder or other geometrical solid in two ways : first by measuring its dimensions; and second by dropping it into a glass graduate partly filled with water and observing the displacement. Do the two results agree ? Which do you consider the more precise method ? 4. Measure a quantity of pure water in two ways: first by placing it in a glass graduate; and second by weighing it on a balance and computing the volume. The weighing may be done in the graduate, which has been weighed beforehand. ON MEASUREMENT 9 5. Lay off on a sheet of smooth paper, with a fine, hard pencil, a Hne of indefinite length, and mark two points on it at random somewhat less than 10 cm. apart. On the straight edge of a card, mark two points as nearly 10 cm. apart as possible. By means of direct comparison with this standard, estimate the length of the first line-segment in centimeters, writing down the result. Next compare the unknown line with a cardboard scale marked off in centimeters but not in millimeters, observing the num- ber of centimeters and estimating the millimeters. Finally compare the same line with a millimeter scale, estimating the tenths of a millimeter. Notice how the three results agree, all being expressed in centimeters. Repeat this several times with different line-segments. 6. Devise and perform exercises, similar to Exer- cise 5, in the measurement of angles, using a large protractor and circular sectors of paper as measuring in- struments. 7. Try measuring short intervals of time to tenths of a second by means of an ordinary watch. In order to test the results, let the period measured be the time of swing of a simple pendulum, and measure by the watch intervals of five, ten, fifteen, twenty and one hundred swings, find- ing the time of a single vibration from each measurement. Do the results agree? Have you any greater confidence in one than in another ? 8. Familiarize yourself thoroughly with the use of as many different kinds of verniers as are available. Before 10 THEORY OF ERRORS AND LEAST SQUARES using the vernier in each case, estimate the fraction of a unit in tenths by the eye. 9. Weigh a small piece of iron by means of a Jolly bal- ance, then on a trip scale, then on an equal arm balance. Compare the results. In which result have you the great- est confidence? Why? 10. Weigh a small object several times, with the high- est degree of precision attainable, on a good balance. The pointer method should be used. Are the results all equal ? CHAPTER II ON THE OCCURRENCE AND GENERAL PROPERTIES OF ERRORS 7. Errors and Residuals. — The term error has so far been used somewhat indefinitely, and it will be necessary, before going further, to explain its exact meaning, as well as that of another term closely connected with it. We have seen that different measurements upon the same quantity generally give different results. These results evidently cannot all be correct, and it is very un- likely that any of them is correct, even to the degree of precision (that is, to the number of decimal places) attain- able with the instruments and method used. The differ- ence between the result of an observation and the true value of the quantity measured is called the error of the observation. In what follows we shall generally denote observations by the symbol s, the quantities upon which they are made by q, and the errors of the observations by X, the latter being defined, as just stated, as the difference X = s - q, (1) which will be positive or negative according as the observa- tion is too large or too small. The student should be care- ful to remember this definition, and to apply it to such illus- 11 12 THEORY OF ERRORS AND LEAST SQUARES trations as the following : If a line is exactly 437 feet long, and the result of a measurement upon it is 436.2 feet, then the error is 436.2 - 437 = - 0.8 ft. While we cannot ordinarily obtain the true value of a measured quantity from one measurement, nor even by averaging many measurements, the method of least squares furnishes us, in the latter case, with a means of calculating what is called the most probable value, which is the closest approximation to the true value that the series of obser- vations is capable of yielding. A familiar illustration is that of a series of direct observations upon a single quan- tity, in which case the most probable value is simply the arithmetical average of the several results. Having obtained the most probable value from a series of observations in the manner hereafter to be explained, if we now subtract it from each measured result, we obtain a series of differences known as the residuals corresponding to the respective observations. The most probable value being denoted by m, and any observation by s, the resid- ual corresponding to s is p = s — m. (2) Thus, the residual bears the same relation to the error that the most probable value bears to the true value. If the num- ber of observations be very large, and the observations be very precise, then the most probable value may be very, very close to, though never equal to, the true value; and in that case the residuals will be equally close to the cor- responding true errors. PROPERTIES OF ERRORS 13 It is worthy of note that, since the true value of a quan- tity in terms of any arbitrarily selected unit is always an incommensurable number, while the most probable value is commensurable, it follows that the error of any observa- tion is incommensurable, while the corresponding residual is commensurable. The true value and the errors are con- sequently forever unknown and figure only in theoretical discussions (with such exceptions as have been noted) ; and we deal in practice only with their close approxi- mations, the most probable value and the residuals. 8. Classification of Errors. — It is now very important to point out that errors of observation may be divided naturally into two distinct classes, whose occurrence, and the methods of dealing with which, are entirely different. First we may consider those errors which arise from causes that continue to operate in the same manner throughout the series of observations, and which may therefore be called persistent or systematic errors. In many cases, persistent errors not only occur in the same manner, but have the same value, throughout the investi- gation, and they may then be called constant errors. The causes of persistent errors, which are often known to the observer and may in many cases be eliminated or avoided by methods presently to be explained, may be, for the most part, looked for under one or another of the following heads. a. Incorrect Instruments. — The instruments or scales used may not be true. For example, if a 100-foot tape 14 THEORY OF ERRORS AND LEAST SQUARES is actually only 99.99 feet in length, every measurement on a line made with that tape will tend to give a result one ten-thousandth too long, no matter how many times the observation is repeated; or if a clock used in scientific work gains one second a day, every measurement on an interval of time made w^ith that clock is just the correspond- ing fraction too long. (In each of these cases, is the error positive or negative ?) b. Imperfect Setting of Scale. — Owing to carelessness or accident, the scale on a measuring instrument, though truly graduated, may be displaced from its proper position by a small amount. This is well illustrated by the mercurial barometer, on which the scale must be adjusted at each reading, to allow for the rise and fall of mercury in the reservoir; and a clock Which, though running at the* proper rate, has been set a little ahead or behind the true standard time, is an analogous case. c. Defective Mechanism. — No instrument is absolutely perfect from a mechanical standpoint, and every instru- ment of precision must be frequently tested if we would rely upon the results of its use. The arms of a balance are never really equal, and, what is worse, they are continually changing their relative length, owing to changes of tem- perature. Nor has it been found possible to construct a clock that will run with absolutely constant rate, even at a constant temperature and in a vacuum. d. Fabe Indicator Settings. — In very delicate instru- ments, such as the balance or the aneroid barometer, the indicator frequently comes to rest, on account of PROPERTIES OF ERRORS 15 friction, in a false position. In the case of the aneroid barometer, it often suffices to tap the dial gently, in order to make the indicator assume its true position. The same may be said of the magnetic compass-needle. e. Knoivn External Disturbances. — It is often the case that persistent errors are introduced by external causes whose nature is well understood, but which cannot be avoided. Thus, a heated body under experimental in- vestigation always radiates some heat, in spite of the most elaborate precautions ; and the length of a measuring rod or tape is certain to vary with changes of temperature. f . Personal Equation and Prejudice. — Every observer exhibits peculiarities or habits of observation which cause him to have a tendency toward persistent error in the same direction. Thus, one observer may continually overestimate in the estimation of tenths, another will under- estimate ; a time observer requires a certain definite interval to respond to a stimulus, that is, to obey a signal of any sort. This unconscious, persistent error on the part of an observer is called his personal equation. Somewhat analogous to personal equation is what may be called prejudice. After an observer has made one measurement of a quantity on a fixed scale, and made the estimation of tenths, there is a natural tendency for him to allow his first estimation to affect the subsequent ones. This difficulty is often met with in the use of the vernier, where it is necessary to judge as to which line coincides most nearly with its fellow on the scale^ The second class of errors referred to at the beginning 16 THEORY OF ERRORS AND LEAST SQUARES of this article comprises those whose causes are temporary, existing through only one observation, and disappearing entirely upon a slight change of conditions. Such errors are not recognizable, and sometimes not even suspected, until their existence is demonstrated by the discrepancies between successive observations when all known disturb- ances have been eliminated. These are known as acci- dental errors. Accidental errors may also be subdivided, as follows. a. Those Due to External Causes. — Accidental errors may result from causes entirely foreign to the observer and of so complex a character as to be incapable of analysis. For example, in sighting a mark with a surveyor's transit, a sudden gust of wind may imperceptibly sway the in- strument for a moment, or someone may, without the ob- server's knowledge, knock against the tripod and jar the telescope slightly out of place. In making delicate mag- netic measurements, such rapid changes as often take place unexpectedly in the earth's magnetic field may momentarily affect the equilibrium of the needle. In sighting at a star with a telescope, currents of air in the upper atmosphere may cause it to waver and appear for a moment to one side of its mean apparent position. In using a balance, the zero of equilibrium may change slightly during the course of a single weighing, owing, perhaps, to an unsuspected fluctuation of tempera- ture. It will thus be seen that observations of all kinds are affected by multitudes of such causes, which are of greater PROPERTIES OF ERRORS 17 or less importance, but which all tend to affect the accuracy of the results. b. Accidental Errors of Judgment. — Aside from per- sonal equation and prejudice, the observer himself is sub- ject to fluctuations of judgment, both as to the adjustment of his instrument and as to the estimation of tenths. An attempt to analyze in detail the causes of these internal tendencies to err in judgment would belong to the realm of psychology ; but we may mention as prominent among them the influences of imperfect vision, optical illusion, inattention and fatigue, the last mentioned cause probably affecting the others in a very large degree. Some of the methods commonly employed in dealing with persistent errors are briefly mentioned in Art. 10. It is, however, the study of accidental errors, and of the laws which are found to govern their occurrence, that constitutes the special office of the method of least squares. 9. Mistakes. — Entirely distinct from errors, in the sense heretofore used, are those inaccuracies which are due purely to carelessness, and which should properly be called mistakes. They consist in such blunders as reading the wrong number on the scale, reading one number and putting another down in the notes, reading a vernier back- ward instead of forward, making a miscount in timing a pendulum, etc. Mistakes are usually easily detected, and there is no remedy except vigilance and careful check- ing. When measurements are made more than once the checking is a simple matter. 18 THEORY OF ERRORS AND LEAST SQUARES 10. General Methods of Eliminating Persistent Errors. — In Art. 8 are enumerated several causes of persistent errors, with illustrations of each. Though their discussion does not properly belong to the general theory of errors, it may not be out of place to describe here some of the methods commonly employed in dealing with them, especially as the theory of errors is frequently applied in the processes of correction here referred to. The treat- ment of the several sources of persistent errors will be taken up in the same order as they are mentioned in Art. 8, and designated by the same letters. a. Incorrect Instruments. Adjustment and Standardiza- tion. — As it is never certain that an instrument measures in true units, it is necessary to test it before relying upon the results of its use. (The tests may in some cases be made long after the measurements.) An instrument may sometimes be adjusted correctly, and remain so; more commonly it gets out of adjustment again, from wear or other causes. Actual adjustment may often be inconvenient or impossible. A more approved practice is standardizor Hon, which will apply to nearly every case. This consists in comparing the instrument with a standard and determining the true value of each of its scale divisions or units, and then, instead of trying to adjust the instrument, simply making the necessary corrections on the observations. (Where standardization extends over a whole scale, it is commonly called calibration.) Thus, the astronomer seldom corrects his clock ; he simply determines its error from the stars at intervals, and thus deduces its error in PROPERTIES OF ERRORS 19 rate, which is all the information needed at any time. Laboratory weights are seldom correct when purchased, and moreover they lose or gain weight by wear or corro- sion ; hence they should be compared from time to time with standards kept for the purpose. Numerous illustra- tions of the kind will occur to the reader. b. Imperfect Setting of Scale. Differential Method. — The error due to imperfect setting of the scale may often be eliminated by the differential method, which consists in reading the position of the indicator when it should be at zero, then again when it is affected by the quantity to be measured, and taking the difference. This method applies only when the scale divisions are equal throughout the scale. The process is one very generally employed, as it has further advantages than the one here stated; very frequently it is the only method practicable. The use of a level and leveling rod in surveying illustrates the latter point, as does almost any kind of comparator ; and when one wishes to weigh a portion of liquid, he must needs subtract the weight of the empty vessel from the weight of the vessel and contained liquid. c. Defective Mechanism. Compensation. — Instru- mental errors may often be made to react against them- selves and automatically disappear. When this can be done, it is by far the best method of elimination. A simple example is the process of " double weighing," in which the effect of inequality in the arms of the balance is removed by weighing with the object first on one pan, then on the other, and taking the mean. (Strictly, the 20 THEORY OF ERRORS AND LEAST SQUARES geometrical mean should be used.) If a spirit level, resting upon an imperfectly adjusted base, be simply reversed, end to end, the half-way point between the two positions of the bubble will indicate its true position as well as if it were in adjustment. The graduated circles used on surveying instruments, spectrometers, and the like, are usually provided with two diametrically opposite verniers, so that the error arising from the vernier system being out of center with the circle itself may disappear on taking the mean of the readings of the two verniers. In using a galvanometer it is well to reverse the current and read the deflection both ways on the scale. An in- teresting application of the method to the elimination of unknown external disturbance is the scheme devised by Rumford for neutralizing the effect of radiation in calori- metric measurements. A preliminary experiment is made to determine by what amount the temperature of the calo- rimeter will be raised ; and then the initial temperature is so adjusted that it is about the same amount beloiv the temperature of the surrounding air at the beginning of the experiment as it is above it at the close, so that practi- cally the same amount of heat is absorbed during the first half of the operation as is radiated during the last half. d. False Indicator Settings. Oscillation. — In cases where the indicator comes to rest in a false position, due to friction, the difficulty may often be removed by not allowing the indicator to come to rest at all, but reading it while still oscillating. This method has the further advantage of saving time in such instruments as the bal- PROPERTIES OF ERRORS 21 ance and undamped galvanometers or magnetometers. In order to compensate for diminishing amplitude, one more reading should be taken at one extreme of the swing than at the other, as in the following balance pointer readings and reduction: Left Right 7.8 13.1 8.0 13.0 8.1 2)26.1 3)23.9 13.05 7.97 13.05 2)21.02 10.51 True reading. This result is much more quickly obtained and more accurate than one obtained by letting the pointer come to rest. e. Theoretical Corrections for Known External Disturb- ances. — When the manner in which external disturbances operate is known, and their magnitude determined, the errors due to them are eliminated by simply applying the computed corrections. The temperature and stretch corrections applied to the steel tape in precise chaining, and the temperature corrections necessary with instru- ments, such as the barometer and pyknometer, depending upon the density of a liquid or the capacity of a hollow vessel, are familiar examples. Instead of employing Rumford's compensation in using the calorimeter, the amount of radiation per minute may be previously noted 22 THEORY OF ERRORS AND LEAST SQUARES and allowed for in the reduction of the results. The re- fraction error in the observed altitude of a star, or in long range leveling, the vacuum correction in weighing, etc., are further familiar examples. It is for the purpose of obtaining data for such corrections that many investiga- tions of the behavior of physical phenomena under varying conditions are carried on ; indeed, this work constitutes a large part of quantitative scientific research. f . Corrections for Personal Equation and Prejudice. — Personal equation may be eliminated, either by deter- mining by means of specially devised experiments what the personal equation of the observer is for a given kind of measurement, or by arranging matters so as to make the personal error act in opposite directions in the two halves of the observation ; or by a very different method, — that of employing a number of different observers on the same measurement, whose errors will tend to compensate in the long run, like accidental errors. The effect of prejudice may often be avoided by altering the conditions. Thus, when repeatedly using the differen- tial method, the whole measurement may be shifted each time to a different part of the scale. The oscillation method is not subject to prejudice, since, though the true reading may be the same in the successive observations, the oscilla- tions approaching it will not be. An experienced observer will not allow prejudice to influence him to any great extent. 11. Exercises Leading to an Understanding of Error Distribution. — Before attempting any introduction to the PROPERTIES OF ERRORS 23 methods of dealing with accidental errors in measurement, it is necessary that the student recognize the existence of a law governing their occurrence, and become to some extent familiar, through experience, with the operations of that law. To this end, it is deemed worth while to introduce at this point a number of laboratory exercises or experiments, in which the phenomenon to be studied is the distribution of errors as governed by the law of chance. The term " laboratory " refers to the method only ; the exercises may be performed at one's study table without any special apparatus. 1. No better analogy to the behavior of accidental errors can be found than in the manner in which shots fired at a target are found to distribute themselves with respect to a point fired at. To illustrate this experi- mentally, take a sheet of ordinary foolscap or other ruled paper and with a black pencil make the ruled line nearest the middle of the sheet heavier than the others, so as to be distinctly visible a few feet away. Lay the paper on a board or smooth book, and place it, face upward, on the floor. Take a rather long pencil lightly between the ex- tended finger-tips of both hands, and standing with the eye directly over the black line on the paper, hold the Fig. 1 24 THEORY OF ERRORS AND LEAST SQUARES pencil, point downward, over the line, and endeavor to drop it so as to strike the line with the descending pencil- point. In other words, make the central line a target; the shots will be self-recorded by the dots on the paper. Take at least a hundred shots in this manner, each time trying with all possible skill to hit the central line. Having done this, prepare another sheet of paper ruled off in a similar manner (ordinary coordinate paper will do) and plot on it a curve whose ordinates represent the relative number of shots found to have struck in each compart- ment of the ruled target and whose abscissas represent the distances of the respective compartments from the central line. In case a shot appears to have struck exactly upon one of the lines, assign it to the compartment on the side toward the center. Can you think of any influence that might, in this ex- periment, be analogous to a persistent error in measure- ment? What effect would it have on the curve? Keep the data for future use. 2. On a sheet of smooth paper, draw a line with a hard, sharp pointed pencil and mark two points on it about a foot apart. The exercise is to measure this line with a metric scale to hundredths of a centimeter, estimating the hundredths as tenths of a millimeter. In order to avoid prejudice, it will be well to place a third point somewhere between the others, and measure the line in two segments, a and b. Now measure a and 6 alternately, using the differential method, until each has been measured, say, a hundred times. Add the corresponding pairs of values PROPERTIES OF ERRORS 25 and record the sums as the measured lengths of the Une. Find the mean of the hundred values to the nearest hun- dredth of a centimeter, and record the departure from it of each of the observations, plus or minus. These de- partures are the residuals of the observations (Art. 7). It will be noticed that a large number of residuals have the same value. Determine how many there are of each value, separating positive from negative, and plot a curve whose abscissas represent the values of the residuals and whose ordinates represent the numbers of residuals having those respective values. A convenient scale should be used : for example, on the abscissas, let 1 cm. represent 0.1 mm. of residual, and on the ordinates, let each residual be represented by a millimeter. Keey the data for future iLse. What change would have to be made in the curve if the abscissas and ordinates were the values and numbers, respectively, of the true errors instead of the residuals, supposing that there is any means of knowing the former ? 3. The preceding exercise may be varied by using, for the measured quantity, an angle of exactly 180°, measuring it in two segments with a protractor to tenths of a degree. In this case the true value, and hence the true errors, are known. Keep the data. 4. Do the curves obtained from the preceding exercises bear any resemblance to each other ? Construct a smooth curve which seems to be typical of them. Does this curve resemble any familiar geometrical form? Plot the curve y = 2"^, taking 10 cm. as the unit for both abscissas 26 THEORY OF ERRORS AND LEAST SQUARES and ordinates and assigning to x the successive values 0, 0.1, 0.2, 0.3, etc., both positive and negative. 5. From the results of the foregoing exercises, does there appear to be any relation connecting the magnitude pf an error with the frequency of its occurrence? Can you assign any reason for such a relation? Do positive errors appear to occur any more frequently, in the long run, than negative errors, or vice versa f 12. Remarks on the Distribution of Errors. — The curve to which the preceding exercises have introduced us is commonly called the probability curve, though a better name would be the curve of departures, as will appear later. Superficially it some- what resembles the " witch," a typical case being shown in Fig. 2. The student must not expect that any curve plotted from the results of such experiments as the foregoing will be smooth and regular, like the curve here shown; actual curves are broken and irregular. But the greater the number of observations or data, the nearer will the actual departure curve assume the smooth, symmetrical form assigned by theory. The results of experiments, as we have seen, and theoreti- cal considerations, as will appear, both point to the follow- FiG. 2 PROPERTIES OF ERRORS 27 ing facts regarding the distribution of accidental errors, all of which may be deduced from an examination of the curve. 1. The frequency with which an accidental error of given magnitude occurs depends upon the magnitude of the error. 2. Large errors occur less frequently than small ones. 3. The error distribution is symmetrical; that is, positive and negative errors of the same magnitude occur with the same frequency. Though these laws do not of course apply absolutely in any one case, yet they express the general tendency of error, and, in fact, the general tendency of all accidental departures from the normal or mean, as, for example, the statures of individual people as compared with the average stature of the race. In theoretical discussions, the number of observations made, or of data considered, is regarded as infinite, and the curve as strictly sym- metrical. In the case of measurements, with which we are here concerned, if the results are affected by persistent error from any source, they will be found to cluster about the theoretical most probable value of the measured quantity instead of the true value, there being now an appreciable difference between the two. The whole curve of errors now becomes a curve of residuals, and is merely shifted a little to one side or the other according as the persistent error is positive or negative. If, for example, in the second exercise of Art. 11, the scale used had its spaces slightly too long, the whole curve would be shifted a little in the 28 THEORY OF ERRORS AND LEAST SQUARES negative direction, simply because each observation tends to undervalue the line measured on account of the defect in the scale. From this consideration it is clear that when, as is really always the case, the true value of the measured quantity is not given by the measurements, a study of the curve of residuals will reveal nothing as to the presence or absence of per- sistent errors. The law of probability of error is con- cerned only with accidental errors, that is, those whose causes are of temporary duration, — the result, as we say, of pure chance. 13. Precision. — On comparison of the results of differ- ent sets of measurements, even upon the same quantity, it is found that the error curve is not of constant form. Every gradation is met with (Fig. 4), from low, flat curves to high, pointed ones. This peculiarity may be observed when we make several series of measurements upon the same quantity by different methods. The variation is easily interpreted. Compare, for example, curves A and D. In the case of A there are nearly as many large errors as small ones. PROPERTIES OF ERRORS 29 For shots fired at a target, this would indicate poor marks- manship or long range ; in measurement, it means random judgment, crude instruments, or circumstances which render the work difficult. In the case of D, on the other hand, the number of large errors is very small, the great body of results being crowded closely about the mean and indicating its position with con- siderable definite- ness. From this it is clear that the form of the error or resid- ual curve depends upon the precision with which the ob- servations have been made. To illustrate what is meant by preci- sion, let two parties of observers each make a set of measurements on tlie dis- tance between two stakes, the one with a ten-foot pole, the other with a steel tape. The most probable value deduced from one set may not differ much from that deduced from the other, but the residual curves plotted from the two sets of results will show considerable difference of precision, mainly on account of the larger number of times that the ten-foot pole must be laid down and its consequent greater liability to error. Fig. 4 30 THEORY OF ERRORS AND LEAST SQUARES The form of the residual curve may therefore be used as a test of the efficiency of an observer, an instrument or a method of measurement. It will be seen later that the same test can be applied by means of mathematical formulas, without the labor of plotting the curve (Chapter VIII). 14. Mathematical Expression of the Law of Error. — The evident existence of some law governing the distribu- tion of errors leads us to inquire what that law is, and whether it is expressible by a simple mathematical rela- tion. Some of the facts concerning the behavior of errors have already been deduced ; but the theoretical expres- sion of the law itself, and even the very language in which' it is expressed, must be reserved until the student has reviewed some of the fundamental principles of the theory of probabilities and has been introduced to some of the special problems in probability upon which the theory of errors is found to depend. The following chapter is, therefore, devoted to this subject. CHAPTER III ON PROBABILITIES 15. Fundamental Principle. — It is a common remark that one thing is more Hkely to happen than another. In speaking thus, one concedes that either of the two events may happen, and attempts no prediction as to which will happen, if either ; yet he recognizes a preponder- ance of the Hkelihood of one event over that of the other. In the kind of magnitude here recognized, that is, likelihood or probability, there is, in the great majority of cases, no means of measuring or giving numerical ex- pression to its relative degrees. It is said that corn growing on low ground is more likely to be caught by frost than that on high ground, but there is no means of telling how many times more likely it is. It is possible, however, to give such a definite meaning to the term probability that the relative probabilities of some simpler events may be calculated and expressed. In framing such a definition, it is necessary to recognize an important principle in the operation of chance, governing the behavior of events whose causes are at least partly manifest, and lying at the foundation of the whole course of reasoning that gives rise to the idea of mathematical probability. 31 32 THEORY OF ERRORS AND LEAST SQUARES The principle is this. // a number of different events are equally possible as regards constant conditions (that is, if there is no persistent reason why one should occur rather than another), and all are repeatedly given oppor- tunity to occur, they will in the long run occur with equal average frequcTicy. The same principle may be ex- pressed by saying that if we observe events occurring with equal frequency, we conclude that the constant conditions under which they occur are uniform. The principle is well illustrated by the throwing of dice. If a die is exactly cubical, of homogeneous material (not " loaded ") and the spots do not shift the center of gravity to one side, and if it be cast a great number of times absolutely at random, each face will come up, on the average, one throw out of six. (Of course these ideal conditions are not realized in practice.) We are so accustomed to the operation of this law of probability in daily experience that it is taken as a matter of course, like the force of gravitation; yet its existence is really a mystery. We are here obliged to admit that there is a law controlling the operations of chance, — the one thing that would seem to obey no law. 16. Definition of Mathematical Probability. — Definite numerical significance may now be given to the probability of occurrence of certain classes of events. If an event may occur in a equally possible ways, and at the same time b equally possible alternatives are presented in all (including the a ways in which the ON PROBABILITIES 33 event may happen), then the probability of the event in question is defined as the ratio V = \ (3) That is, there are a chances favoring the event out of a total of b possible chances ; and according to the principle set forth above, if a great number of trials are made, the event does happen, on the average, a times out of b. As an example, let us express the probability of draw- ing an ace from a deck of fifty-two playing cards, the draw- ing being done absolutely at random. Any one of the fifty-two cards may be drawn, so that the total num- ber of alternatives is fifty-two. An ace may, however, be drawn in only four ways, viz., by drawing the ace of spades, the ace of clubs, the ace of hearts or the ace of diamonds. Here, then, & = 52, a = 4, and the probability of drawing an ace is ^^, or -^-^. What would be the probability of drawing a red ace? Of drawing the axie of diaTnonds ? All problems in probability may be solved by the appli- cation of the definition expressed in equation (3). But such direct application would be very diflOicult in the more complicated cases, and special rules and formulas are therefore to be devised wliich, when properly classified and applied, greatly simplify such problems. From the definition, it follows that probability is a purely numerical ratio, and depends upon no unit of measure. Moreover, this ratio cannot exceed unity. The probability unity would denote certainty, since if an 34 THEORY OF ERRORS AND LEAST SQUARES event may happen in n ways, and only n alternatives are possible, the event must happen. From this it follows that if the probability of an event is p, the probability of its failure to happen is p' = 1 - p. (4) For, if the event can happen in a ways out of b, it can fail to happen inb — a ways out of b, the probability of failure therefore being b—a -, a -, — =i--=i-p. More generally, the sum of the probabiHties of all possible alternatives is unity. The probability zero, on the other hand, implies im- possibility. It may be interpreted as meaning that there is no way for the event to happen, i.e., a = 0; or in cases where the total number of alternatives is infinite, or at least extremely large, while the event in question may happen in only a very few ways, the zero or infinitely small probability denotes impossibility or at most only extremely remote possibility. But the distinction be- tween absolute impossibility and the case in which the possibility is only remote is of some importance, as will be seen, in the theoretical discussion of the distribution of errors. 17. Permutations. — The solution of problems in prob- ability involves the determination of the number of ways in which an event can occur, as well as the total number of possible alternatives. In very simple cases this may be ON PROBABILITIES 35 done by inspection. For example, if one is expecting the arrival of three different persons, A, B, C, it is easy to determine the probability of their coming in the order named. There are obviously six different orders in which they may come ; namely, ABC, ACB, BAC, BCA, CAB, CBA. The probability of their coming in the order ABC is therefore J. But let there be a hundred persons instead of three, and the number of orders becomes so enormous as to be unmanageable by inspection. We must then resort to the use of general formulas. The linear permutations of a number of things are the dif- ferent ways in which the things may be arranged in a row, or in which they may occur in order of time. There are, for example, six linear permutations of the letters A, B, C. There is a general expression for the number of permuta- tions of Q different things, derived easily by the following reasoning. Of one thing, there is evidently but one per- mutation. Of two things, since either may come first, there are two permutations. Of three things, any one may come first, and with a given one coming first, there are two arrangements of the two remaining; therefore the number of permutations of three things is 3 X 2 = 6. For four things, by the same reasoning, the number is 4 X 3 X 2 = 24. And in general, the number of per- mutations of Q things is Pq = Q(Q-1)(Q-2)-3.2.1=Q! (5) We have here assumed that none of the Q things are duplicates. Let us now take a case where there are 36 THEORY OF ERRORS AND LEAST SQUARES duplicates, as in the group of letters AABBBCCCC. If we distinguish between the different A's, etc., the case is, of course, the same as if the letters were all different. But if we consider one A, for example, the same as another, and permute without regard to which of them is being used in a particular place, the number of permutations is less. It will be easy for the student to show as an exercise that if there are, in a number of things, m of one kind, n of another, r of another, s of another, etc., the total number being Q = m -]- n + r -\- s -\- •••, the number of distinguishable permutations of the Q things is -, ml nl rl Si ••• Thus for the above set of nine letters, of which two are A's, three B's and four C's, the number is p (2.3.4)^ — ^J — = 1260. 2!3!4! If there is only one thing of a kind in the group, so that m, n, .' . . are each unity, (6) becomes equivalent to (5). 18. Combinations. — The different groups which can be formed from a number of things, taken so many at a time, are called combinations. The different combinations of the three letters A, B, C taken two at a time are AB, AC, BC. If we further take into account the possible permuta- tions of each combination, we have what may be called the permuted combinations of the series of things considered. ON PROBABILITIES 37 Thus the permuted combinations of A, B, C are AB, BA, AC, CA, BC, CB. It is easier to derive first the general formula for the number of permuted combinations. Let the number of permuted combinations of Q things taken n at a time be designated by the symbol PCq^^\ If they are taken two at a time, any one of the Q things may be taken as the first, and any one of the Q — I remaining things may be taken as the second, so that If taken by threes, any one of the Q (Q — 1) permuted combinations of two each may constitute the first two, followed by any one of the Q — 2 remaining things as the third. Then By continuing the same reasoning until there are n things taken at a time, we readily deduce PCq^-^=Q{Q -DiQ- 2) ... to n factors. (7) If n = Q, this becomes identical with (5), since all the things are permuted at once. To express now the number of combinations of Q things taken n at a time, without regard to their arrangement, it is necessary only to note that the PCq^""^ permuted combina- tions include not merely those made up of different things, but all the permutations of each of the groups of n different things. Since n things are permuted in n ! different ways (5), there are only — - as many combinations as permuted nl 38 THEORY OF ERRORS AND LEAST SQUARES combinations. That is, ^ M^ QjQ - 1) ••• to n factors .^. n ! As an illustration of this problem, let us find how many different hands at whist, each made up of thirteen cards, could be drawn from a pack of fifty-two cards. Here Q = 52, n = 13, and the solution is ^^^(13) ^ 52 -5^1 -50 ♦♦^■40 ^ 635^013,559,600. Then the probability of drawing any one specified hand is, by definition, the exceedingly small reciprocal of this number. As a final problem in combinations, let there be s series of things, the number of things in the respective series being Qi, Q2, •••, §«; to determine how many different combinations can be formed by taking one thing from each series. . The number of combinations of two each which can thus be formed from the first two series is Q1Q2, since each of the Qi things in the first series can be successively com- bined with each of the Q2 things in the second. Bringing in now the third series, each of the Q1Q2 combinations just considered may be combined with each of the Q3 members of the third series, making Q1Q2Q3 combinations ; and so on. Clearly, then, the number of combinations that can be so formed from the s series is the product .CQ,..Q, = (iiQ2Q3-Q.^ (9) ON PROBABILITIES 39 For example, let there be three series of letters : A, B, C, A2 B2 C2 A, B, A, The number of combinations of the form ABC that can be selected from them is 4 X 3 X 2 = 24. Let the stu- dent write these combinations. 19. Probability of Either of Two or More Events. — If the probability of an event A is pa, that of an event B is Pby that of an event C is pc, etc., then it is easy to show- that the probability that one or another of these events will happen is pa -\- Pb + Pc -\- -", it being understood that only one of these events can happen. For, suppose the event A may happen in a ways, the event B in 6 ways, etc., and that the total number of alternatives is T. (In general, T will be greater than the sum of a, b, etc. ; that is, it is not necessary that any one of the events A, B, etc. shall happen.) Then by definition, the probabilities of the respective events are a b . Va^-^, P6 = ^. etc. If we designate by X the event of some one of the events A, B, etc. happening, without specifying which, then, since the number of ways in which X can occur is a + 6 + ••*, the probability of X is Px= rr =Va-\-Vh-\- •••• (10) 40 THEORY OF ERRORS AND LEAST SQUARES As an example, let there be in a bag three balls of iron, two of glass, five of wood, seven of lead, six of rubber, one of ivory and four of copper, and let one be drawn at random. The probability that a metal ball will be drawn is then A H" A + 2*? = i* since a metal ball is drawn if the re- sult be an iron ball, a lead ball or a copper ball. This principle of additive probabilities for alternative events is made use of in estimating premiums on so-called " joint " life insurance policies. 20. Probability of the Concurrence of Independent Events. — Quite a different problem is that of finding the probability that all of a specified set of independent events shall occur. As before, designate the respective events by A, B, C, etc., their respective separate probabilities by Pa, Pb, etc. ; and designate the event of their all occurring by Z. Suppose the event A may occur independently in a ways out of a alternatives, B in 6 ways out of /3 alter- natives, etc., so that a h . Pa = -, pb = -, etc. a p It is of course understood that when all the events A, B, C, etc., are given opportunity to happen, some one of the a alternatives connected with A will happen, some one of the j8 alternatives connected with B ivill happen, etc., but that only one of each can happen. The total number of possible outcomes is therefore the number of combinations that can be formed by selecting one from each group of alternatives, namely, the product ajSy ••• (9). Likewise, ON PROBABILITIES 41 the number of different ways in which the events A, B, etc., can all occur is the product abc •••. It follows that the probability of all occurring, that is, the probability of the event Z, is abc '"abc .^ ^. Vz = — = - '-'-•" =PaPbPc •••• (11) apy ••• a p y That is to say, the probability of the concurrence of two or more independent events is the product of the probabilities of the respective events considered separately. This product is of course less than any one of its factors. To make the meaning of this clear, suppose that it is known that a person A will spend five hours in a certain place between 6 a.m. and 6 p.m., and that another person B will spend three hours there during the same interval, but nothing is known as to when these hours will be. If we visit the place at any random moment, the probability of finding A there at that moment is i\ ; the probability of finding B there at that moment is y2 . Then the probability of finding them both there at that moment is i\ X i% = ^\. But the probability of finding either A or B there is ■^2 + -12 = I- Let the student analyze this problem more closely, showing how the values stated for the probabilities can be deduced from the definition of probability. 21. The Coin Problem. — Suppose that the result of an experiment may be either one of two things, A and B^ which are equally likely to occur, and that the result must be one or the other, but cannot be both. The probability of either result is then J. Let us determine what is the 42 THEORY OF ERRORS AND LEAST SQUARES probability, if the experiment be performed Q times, that it will result n times one way and Q — n times the other. A coin tossed at random illustrates the problem ; for example, if it be tossed a hundred times, what is the prob- ability that it will turn up heads thirtj^-eight times and tails sixty-two times? Here Q = 100, n = 38 (or 62). The required probability is a function of n ; and, further- more, it is evidently the same function oi Q — n that it is of n. The first thing to determine is, in how many ways the result A may happen n times out of Q. In 100 throws of the coin, heads may come up 38 times and tails 62 times in a large variety of ways : for example, 1 H., 2 T., 37 H. and 60 T., in order, would fulfill the condition ; or, equally well, 8 H., 5 T., 30 H., and 57 T. The number of ways in which A may happen n times and B, Q — n times is readily seen to be equal to the number of distinguishable permutations of Q things, n being of one kind and Q — n of the other (Art. 17), which is Pq^^^^-^^= . ,^' ., (12) 71 ! {Q-n)l Or, it is equal to the number of combinations of Q things taken n at a time, since out of the totality of Q events, the n events A may be selected wherever desired. Hence another expression for the required number is equation (8), which the student may readily show to be equivalent to (12). We shall use equation (12). Next we must determine the total number of possible ON PROBABILITIES 43 alternatives. This may be done by adding together the values of the expression (12) obtained by giving n all integral values from to Q. These are tabulated below. n 1 p^(«, Q-n) 1 Q QXQ-i) Q(Q 2! -iXQ-2) 3! QiQ-i) 2! Q 1 Q-2 Q-i Q The expressions obtained for Pq(^'LE OF LEAST SQUARES 25. Analogy of Error Distribution to Coin Problem. — It was pointed out in Art. 5 that an error in measurement is the resultant of innumerable small disturbances of different kinds, the presence of many of which may not be even suspected. These disturbances operate, some in one way, some in the other ; that is, some tend to produce positive error and some negative. The resultant error depends on the 1-elation of the number of positive disturb- ances to the number of negative disturbances. If nearly all are positive, the error will be positive and large; if nearly all are negative, a large negative error will result ; while if about the same number are positive as negative, the error will be small. This does not imply that the dis- turbances are all of the same magnitude. By way of illus- tration, suppose we select from a sand-heap, at random, a thousand grains of sand, and put eight hundred of them on the left pan of a balance and two hundred on the other. There is hardly a remote possibility that the former will not very largely overbalance the latter. But if we put five hundred on each pan, there will be little preponder- ance one way or the other. And this does not imply, by E 49 50 THEORY OF ERRORS AND LEAST SQUARES any means, that the grains are all equal in weight ; some individual particles may be ten times heavier than others. Now there is a remarkable and useful analogy between the theory of error distribution and the so-called coin problem (Art. 21), an analogy that the student has no doubt already observed. It is easily deduced that the most probable result of a number of throws of a coin is that they will be half heads and half tails. In general, this normal result will be departed from in greater or less degree, so that in one hundred throws we frequently ob- tain fifty-five heads and forty-five tails, or less frequently, sixty heads and forty tails, etc. This departure from the normal or most probable result may be looked upon as a sort of error. Like an error in measurement, it is com- plex in character, depending upon the result of each in- dividual throw. Each throw, head or tail, affects the final outcome one way or the other, just as each small disturbance, positive or negative, affects the result of an observation in measurement. A little consideration of the two cases will bring out their analogy quite clearly. We are therefore justified in assuming that the proba- bility of the occurrence of an error is a function of the magnitude of the error in much the same manner as the probability of a departure from the half-and-half result in tossing the coin is a function of the extent of the de- parture. It is, in fact, upon this line of reasoning that Hagen's deduction of the error equation is based. The deduction is, however, rather cumbersome, and we shall follow instead the more elegant method due to Gauss. THE ERROR EQUATION 51 It may be remarked here that the theory of departures is a very general one and finds apphcation in a large variety of problems of common experience, such as the distribution of shots on a target and the distribution of given charac- teristics among the members of a biological group. 26. The Most Probable Value from a Series of Direct Measurements. The Arithmetical Mean. — If a series of measurements be made upon a single quantity under as nearly constant conditions as possible, the result is, in general, a series of different values, each approximating the true value of the measured quantity. No one of them is the true value, however, and it now becomes a matter of judgment to select, from all possible values, such a one as will make the actual distribution of the results appear most natural. An analogous case would be this : Suppose that after all the shots had been fired at the target in the first exercise of Art. 11, the central line aimed at were erased, and we were required, from the given distribution of the shots, to judge as to where the line had been ; we could do no better than to select a position that, from the concentration of shots about it and their symmetry with respect to it, seems to be the most probable one. Likewise, in a series of measurements, we are aiming at a true value, the most probable location of which can only be estimated by an examination of the distributed results. The symmetry of the distribution of errors in cases where the true value is known, as also in the analogous coin and target problems, leads at once to the common axiom of 52 THEORY OF ERRORS AND LEAST SQUARES experience, that the best value to adopt in the ease of a series of direct observations on a single quantity is the arithmetical mean or average of the observations. If the several measured results be designated by Si, S2, "',Sn, and their mean be m, then the residuals (Art. 7) are re- spectively pi = si — m, P2 = S2 — m, Pn = Sn - m. Adding these we obtain 2/3 = 25 - nm = 0, . (15) which expresses the fact that the arithmetical mean of the results is the value with respect to which they are sym- metrically placed, the algebraic sum of the differences being then equal to zero ; and that therefore this mean is the most probable value that can be assumed. 27. Gauss's Deduction of the Error Equation. — Let q represent the unknown true value of a quantity and let a series of n measurements be made upon it, the number n being supposed very large. Let the errors arising from the respective measurements be Xi, X2, •••, Xn. It has been seen that the probability of the occurrence of an error is some sort of inverse function of its magnitude. Designating the probabilities of these respective errors Kv 2/ij 2/2> •••> Vnt this fact may be expressed by the equations THE ERROR EQUATION 53 It is the form of this function f(x) that we are seeking to determine. Now, as above noted, we do not know the true value q of the observed quantity, and therefore we do not know the true errors x. We may however assume various tenta- tive values for q and study the resulting tentative systems of errors, particularly with a view to selecting that one which seems most naturally distributed, in accordance with the notions of error distribution that experience has taught us. In this sense, therefore, we may think of q and the errors x as variables subject to our control, and the probabilities y will then vary accordingly. With this understanding, then, we are seeking to find that system of values for the a:'s which, as a whole, has the greatest probability. If the outcome of a series of measurements be the sys- tem of errors Xi, X2, •••, Xn, this result may be looked upon as the concurrence of n independent events, each of which is the obtaining of one of the errors x. Then according to Art. 20, the probability of this outcome, designated by y, is the product of the probabilities of the separate errors, namely Y = yiy2 - 2/n = /fe) -/fe) -fixn). (16) In order, therefore, that the system of a:'s shall have the greatest probability, as required, the value assumed for q 54 THEORY OF ERRORS AND LEAST SQUARES should be such that the expression F is a maximum; which condition will be attained when dY ^ = 0. (17) dq Differentiating (16) dY^^_ ^ dJM_^_y_ . dfjx,) ^ ^^ dq f{xi) dq fix^) dq Y . df{Xn) _ Q f{xn) dq or y\df{x,) ^df{x2) I ,, J dfjxjl dqLfiXi) f(X2) f{Xn)J = T\.d log Rxi) +d log f{x2) + ... + ^ log f(Xn)] = 0. dq Now let d \ogf{x) = (f)(x)dx, where {xi)^ + (x2)^+^- + cf>M^=0. (18) dq dq dq If the results of the respective n measurements on q be designated by ^i, ^2, •••, ^n, having definite, fixed values, then the errors x are (Art. 7) Xi = si- g, X2=S2- q, THE ERROR EQUATION 55 from which, at once, dq dq dq (18) then reduces to 0(^1) + 0(^2) 4- • • • + {xj = 0. (20) We already understand enough of the law of error dis- tribution to know that when the number of observations is very large, the number of positive errors of given magni- tude about equals the number of negative errors of the same magnitude, and that therefore the algebraic sum of the errors is approximately zero. Since in our theoretical discussion the number of observations is indefinitely large, we may write, therefore, as another condition fulfilled by the errors, Xi-\-X2-{- ••• -\-Xn = 0. (21) It now remains to deduce from the two equations (20) and (21) the form of the function , from which the original function / may then be obtained. It is not difficult to see that the equations are satisfied if {X2) = KX2, ' 0(a:„) = Kxr., where X is a constant. A mathematical proof that this is the necessary relation is given in Note A, Appendix, being omitted here to avoid distracting attention from the 56 THEORY OF ERRORS AND LEAST SQUARES main problem. We may write, then, {x) = Kx; (22) or since {x)dx = d log f{x) = d log y, d\og y = Kxdx. Integrating, log 1/ = i Kx^ + c', or y = e^^^^+<^. (23) This is one form of the error equation. The expression may, however, be so modified as to ex- hibit the relation to better advantage. We have seen that the larger the error, the less likely it is to occur : the larger x is, the smaller is y. Clearly, then, K must be a negative quantity. Replacing ^ i^ by — h^, and e*^ by the constant c, the equation assumes the more usual and more useful form y = ce-^''^\ (24) This is "the most important equation in the theory of errors, and should be committed to memory. 28. Discussion of the Error Equation. — It will be interesting to examine equation (24) to see how closely the law of error thereby expressed agrees with the conclu- sions already reached. The bilateral symmetry of the function y is evident from the occurrence of x in the second degree only. This indicates the equal probability of positive and negative errors of the same magnitude. The function approaches THE ERROR EQUATION 57 zero as x increases in magnitude ; which means that very great errors are extremely improbable. The derivatives of the function are dy dx = -2ch^xe-^'-\ dx^ (25) (26) From these, since -^ = 0, — ^ < when a: = 0, there is a dx dx^ maximum value of y when a: = ; that is, the error zero has the greatest probability. The curve shown in Fig. 5 represents the function, and has some interesting properties. Its symmetry, asymptotic character and central maxi- mum merely illus- trate what has just been deduced from the equation. The Y intercept, or maxi- mum ordinate, is the quantity c, since y = c d'y Fig. 5 when X = 0. If we put ^ equal to zero, which is the dx^ condition for points of inflection, (26) gives 1 - 2 hV= 0, 1 Xi = ± h^J2 (27) 58 THEORY OF ERRORS AND LEAST SQUARES This is the distance OD or OD', corresponding to the points of inflection P and P', The ordinate of these points is, by substitution, 2/i = + -^, (28) and is therefore proportional to c. The quantity c represents the probabiUty of the error zero. Now the probabihty of any given error a: is a function of both c and h, since it changes if we change either c or h. It would thus appear that c and h have something to do with the precision of the measurements, and that they are therefore connected with each other. We shall see later (Art. 54) that this is the case, and also that there is still another factor in the probability of a given error, depending upon the value of the smallest scale in- terval in terms of which the measurements are expressed. 29. The Principle of Least Squares in its Simplest Form. — We are now in position to make an introductory state- ment of the important principle which gives this branch of science its name, — the principle of least squares. Be- fore we are through with the theory of errors, the principle will have been stated several times in successively more complicated forms, as the problems to which it is applied become more and more general. So far we have been considering only the simplest case, namely, that of ob- servations of equal precision upon a single quantity; and while for this case the method of deducing the most probable value is clear without reference to the principle THE ERROR EQUATION 59 of least squares, still it will be interesting and instructive to observe how the assumption of the arithmetical mean as the most probable value may be shown to be in accord- ance with that principle in the simple form here stated. The simple form of the principle referred to is as follows : The most probable value of a measured quantity that can be deduced from a series of direct observations, made with equal care and skill, is that for which the sum of the squares of the residuals is a minimum. The law governing the distribution of errors has already been deduced theoretically, and the experience of number- less experimenters testifies to its truth. We have there- fore a right to expect that, when we have made a long series of measurements upon a single quantity, our observations will have grouped themselves around the true value in a manner approximately consistent with the error equation (24). Then it is logical for us to assume a value for the measured quantity, such that the results of the measure- ments will be so grouped with respect to it. This is the so-called most probable value, and it is the office of the prin- ciple of least squares, in any case, to point out the way of arriving at it. Let the results of the n observations be ^i, ^2, •••, Sn- Then if we designate the most probable value sought by m, there will arise a corresponding series of residuals Ph P2, •'•, Pn, each of which is found by subtracting m from the corresponding observation s (Art. 7). If m be properly chosen, the residuals derived from it will, like true errors, be found to be distributed in accordance 60 THEORY OF ERRORS AND LEAST SQUARES with the exponential law of error probability (24), so that the probabilities of the respective residuals are 2/2 = ce-^'^, 2/n = ce-^V. The probability of the simultaneous occurrence of the assumed system of residuals is then (Art. 27) Y = i/i?/2 ••• pn = c^g-^'^('"^+''^+ - +V). (29) Now if m, and consequently the residuals p, are to be so chosen that the resulting distribution is the most probable one in accordance with the law of error, these quantities must be given such values that the probability Y is as great as possible. But this will be secured, evidently, by making Pi^ + Pi" + ••• + /°n^ as small as possible, as will be seen at once from (29) . That is to say, m should be so chosen that 2/3^ is a minimum, which is the principle of least squares stated above. In order to find what this required value of m is, we may write ^p^={s^-my + {s^-my+ ... -V{Sn-my _._ =a minimum. Hence -^2f)2 = _2[(5i-m) + (^2-m)+ ... +(5„-m)]=0, dm or reducing, m = '^i+'^2+ - +^n ^ (3Q) n which is simply the arithmetical mean of the observations s. THE ERROR EQUATION 61 EXERCISES 30. 1. Show how, in the first experimental exercise of Art. 11, the errors of aim may be due to many minor causes, enumerating as many such possible causes as you can think of. 2. Find the algebraic sum of the errors of measurement in the third exercise of Art. 11 ; also the algebraic sum of the residuals. 3. Plot the curve y = ce~~^^^\ giving the value unity to each of the constants c and L This may be done by use of logarithms {e = 2.718 •••)• Let the unit abscissa be 10 squares and the unit ordinate 50 squares. Compare with the error curves obtained from Exercises 1, 2 and 3 of Art. 11, and with the coin problem curve obtained in Art. 22. 4. Draw a smooth, symmetrical curve which follows as closely as possible the irregular curve obtained in Ex. 3, Art. 11, making it conform to the known prop- erties of the law of error as represented in Fig. 5. From this curve, determine the relative probabilities of the errors of magnitude 0°.l, 0°.2, etc., out to 5°. By locating the points of inflection, find an approximate numerical value for h, 5. Plot the curve represented by 2/ = (2 - x)2+ (3 - xy + (4 - xy + (5 - xY + (6 - xy. Has it a minimum point ? What does this illustrate ? 62 THEORY OF ERRORS AND LEAST SQUARES 6. The number of rays in the lower valve of a certain species of Atlantic mollusk was counted in 508 individual cases. Of these, 1 had 14 rays, 8 had 15 rays, 63 had 16 rays, 154 had 17 rays, 164 had 18 rays, 96 had 19 rays, 20 had 20 rays, 2 had 21 rays. Plot a curve in which abscissas represent the number of rays and ordinates the corresponding number of individuals. What is the probability that two of these mollusks, picked up at random, will each have exactly fifteen rays? (Data from Davenport, Statistical Methods.) 7. Tests were made on fifty schoolboys of equal age to ascertain strength of grip. The following data (Whipple, Manual of Mental and Physical Tests) are in hundreds of grams. 158 210 248 296 348 175 220 262 301 350 193 225 262 310 353 197 225 267 313 375 197 225 269 315 375 200 226 270 320 403 205 235 273 323 430 206 244 280 325 440 208 244 290 330 440 210 245 294 346 508 THE ERROR EQUATION 63 Arrange a suitable curve showing departures from the average or normal strength from these data. 8. About two hundred individuals were tested at the University of Iowa for accuracy of tone perception, the results being expressed by the number of vibrations in the departure from the true tone (international A, 435 per sec.) that the individual could distinguish. The data are expressed in per cent. Departure, Vib. Per Cent. 1 13.8 2 24.0 3 25.5 5 17.3 8 7.3 12 3.2 17 1.6 23 2.7 30 or over 4.6 Plot a curve representing this distribution, and discuss its form. 9. Out of a class of exactly 100 college freshmen, the age 1 was 16, 2 was 22, 12 was 17, 1 was 23, 31 was 18, was 24, 22 was 19, was 25, 18 was 20, 1 was 26. 12 was 21, Plot curve and discuss its form. 64 THEORY OF ERRORS AND LEAST SQUARES 10. Out of over 100,000 public school grades examined by Mr. L. L. Fishwild, 1 per cent, were 50, 1 per cent, were 55, 2 per cent, were 60, 2 per cent, were 65, 5 per cent, were 70, 6 per cent, were 75, 13 per cent, were 80, 13 per cent, were 85, 25 per cent, were 90, 23 per cent, were 95, 9 per cent, were 100. Plot curve and discuss its form. CHAPTER V ON THE ADJUSTMENT OF INDIRECT OBSERVATIONS 31. Observations on Functions of a Single Quantity. — It has been pointed out that measurements are seldom made directly upon the quantities whose values are sought, but are usually made upon functions of them, or functions involving them with other unknown quan- tities. The former case being the simpler, we shall consider it first. As a specific problem, let a number of measurements be made upon the diameter of a circle, with the object of determining its area. That is, the quantity really sought is the area, but the direct measurements are made upon the diameter, a function of the area. Supposing the observa- tions to be all made in the same manner, the question arises, what is the most probable value of the area ? Is it the arithmetical mean of the areas computed from the separate measurements on the diameter, or is it the area determined by taking, as the diameter, the mean of the measurements upon it? The two are of course not the same. This question may be answered by the following general deduction. The quantity whose most probable value is F 65 66 THEORY OF ERRORS AND LEAST SQUARES sought being q, and the function of it, upon which the ob- servations s are directly made, being /(g), there arise the following approximate statements, known as observation equations, each of which represents one of the n measure- ments : fiq) = si, fiq) = S2, f{q) = Sn. (31) Si, S2, ••*, Sn are the results of readings on some sort of scale or measuring instrument applied to the function directly measured. The errors of the observations are represented by Si—f{q), etc., but are not determinate. It is the most probable value of q that we are seeking, and if this be repre- sented by m, the residuals of the n observations are Pi = si -f{m), pi = S2 -/(m), Pn =Sn- f{m). (32) There is no reason why the principle of least squares should not apply to this case as well as to the case of direct measurements, since the law of error distribution, or the " law of departures," is universal in its scope. As relating to this sort of measurements, then, the principle of least squares takes the following form : The most probable value of an unknown quantity that can be derived from a set of observations upon one of its functions is that for which ADJUSTMENT OF OBSERVATIONS 67 the sum of the squares of the residuals arising from these observations is a minimum. The sum above referred to is expressed by ^P'=[si-f(mW+[s2-fim)f+ '" +K-/(m)P, (33) in which m may be regarded as a variable whose value is to be so adjusted as to render 2/3^ a minimum. This con- dition requires that dm or, differentiating (33), - 2[25-n/(m)]-^=0, • dm /(m)=-'. (34) n Therefore m, the most probable value of q, is that value whose /-function is the mean of the observations upon Thus, the most probable value of the area of a circle, as determined from measurements upon the diameter, is - times the square of the arithmetical mean of the results 4 of those measurements. A multitude of other illustrations of this principle will occur to any one familiar with such work. 32. Observation Equations for More Than One Un- known Quantity. — Very frequently, in an experimental research, occasion arises to determine, not merely one, but 68 THEORY OF ERRORS AND LEAST SQUARES several, unknown quantities or constants which are so involved with each other and with the phenomena directly observed as to render their separate measurement im- possible. The following illustrations will make this clear. In the use of the zenith telescope for finding the latitude of a station, the quantities first sought are the zenith dis- tances of two stars selected for the purpose. The sum of the zenith distances is equal to their difference in declina- tion, as given in the star catalogues, and therefore depends upon the results of many very precise measurements made with other instruments at fixed observatories. The difference of the zenith distances is measured by means of the micrometer belonging to the zenith telescope, as the instrument is rotated from north to south about the vertical. In this way, neither zenith distance is separately determined, both being found by the simulta- neous solution of the equations arising from the above observations. Again, it is desired to find the relative proportions of sodium chloride (NaCl) and potassium chloride (KCl) in a mixture of the two salts. Or specifically, in a given specimen of the dry mixture, to find the number of grams, X, of sodium chloride and the number, y, of potassium chloride. First let the sample be weighed, with the result Si. Then , X -]-y = si. The sample is now dissolved and the chlorine precipitated with silver nitrate (AgNOs), and the total amount of chlorine present calculated by weighing the precipitated ADJUSTMENT OF OBSERVATIONS 69 silver chloride (AgCl). Denote the total chlorine by ^2. Now, sodium chloride is 0.6123 chlorine, and potassium chloride is 0.4754 chlorine. Hence in x grams of sodium chloride and y grams of potassium chloride, the total chlorine is o.6123 x + 0.4754 y = ^2, which furnishes the second observation equation necessary for obtaining x and y. This is another instance in which neither of the unknown quantities is measured separately. Quite often only certain ones of the unknown quantities are really desired, the others being merely troublesome corrections or instrumental constants which must be de- termined or eliminated. The method of procedure, how- ever, is the same in this as in any other case. 33. More Observations than Quantities. Normal Equations. — In the illustrations of the preceding article there were, in each case, two unknowns, and two inde- pendent observations were necessary to determine them. By independent observations are meant observations made on a different principle, or under such different con- ditions that the resulting observation equations will have different coefficients and not merely different absolute terms. To repeat the process of measuring the sum of two unknowns, without attempting to find some other relation between them (as, for example, their difference or their product), would give no information as to the separate values of the unknowns. And, in general, the de- termination of / unknown quantities requires a knowledge of / independent and consistent relations between them. 70 THEORY OF ERRORS AND LEAST SQUARES If measurements could be made without error, the solu- tion of the / independent observation equations formed from such measurements would give us the values of the / unknowns exactly; more than / measurements would be superfluous. But, as in the simpler case of a single un- known, the existence of accidental errors makes it desir- able to get as many observations as possible, and to devise some means of averaging them so as to find the most probable value of each of the unknowns. This prob- lem is the most important that arises in least squares. Let there be n observations upon functions of the I un- known connected quantities qi, q2,"', Qi {n>l), and let the series of resulting observation equations be repre- sented by h (qu qi, qi) = si, qi) = ^2, fn{qi, ?2, •••, qi) =Sn. (35) Here, as in the simpler cases, there are errors and residuals obeying the same law of error distribution set forth in the error equation. We are seeking to obtain the most probable values, mi, m2, •••, nii, of the unknown (and unknowable) quantities q that the observations will furnish, and when these are found, the n residuals will be given by Pi = si-fi (mi, m2, •••, TUi), P2 = S2-f2 {rni, rui, •••, m,), ^ P« = Sn-fn('rrii,m2, •-, m,). ADJUSTMENT OF OBSERVATIONS 71 The principle of least squares may now be slightly modi- fied in wording to fit this case, thus : The most probable values of unknown quantities connected by observation equa- tions are those which will render the sum of the squares of the residuals arising from the observation equations a minimum. It is possible, through an application of this principle, to reduce the n residual equations (36) to a number /, equal to the number of unknowns, which can then be solved for the most probable values m. The process may be regarded as finding from the n observation equa- tions (35) a set of / most probable equations whose solution will give the most probable values of the unknowns q. From the principle of least squares, the sum pi^ + p2^ + ••• +Pn^ must be a minimum, and in order that mi, ?7i2, •••, mi may be so selected that this will be the case, the first partial derivative of this S/a^ with respect to each of those quantities must be zero. (See any calculus.) That is, _d_ dmi ^[s-f (mi, m2, •••, mi)Y = 0, dm2 ^mi 1 (37) The equations (37) resulting from these differentiations are the most probable or normal equations required, and 72 THEORY OF ERRORS AND LEAST SQUARES being / in number, will yield, on solution, the most probable values mi, mz, •••, mi, which are sought. Equation (34) is a normal equation, containing only one unknown, m. 34. Reduction of Observation Equations of the First Degree. — In nearly all cases in which the method of least squares is used in the reduction of observations in accord- ance with the foregoing theory, the observation equations are either all cf the first degree, or they may, by suitable substitutions, be replaced by equivalent observation equations which are of the first degree. The mathemati- cal operations required in finding the normal equations are then comparatively simple, and can be performed with- out any knowledge of calculus. Let the n first degree observation equations upon the I quantities q (corresponding to (35)) be as follows : (38) tti^i + biq2 + Ciqs + h nqi = Si, a^qi + hqi + ^273 H h r^qi = s^, anqi + M2 + ^n^sH h Tnqi = Sn. The residuals will then be Pi = si- {aiini + biJUi + ••• + nmi), P2 = S2 — {a^mi + 627772 + • • • + r2mi ), ). Pn = Sn- {cinrni + Knii + ••• + rnTTii (39) Only one term in each of these expressions contains mi ; denote the balance of the expression in each case by a single ADJUSTMENT OF OBSERVATIONS 73 letter, as B. Then pi = — airrii + Bi, etc., and Z/)2 = ( - aivii + BiY +•••+(- a„mi + B^Y. with respect to mi, as pe: 2/^2 = — 2 tti ( — aimi + Bi) Differentiating with respect to mi, as per equations (37), d drrii -2an{- anTThi + Bn) = 0. Or dividing by 2 and remembering that — am + B = p in each term, - aipi - aiPi anpn = 0. (40) This is the normal equation pertaining to mi, and corre- sponds to the first of equations (37) . This result may be directly obtained by multiplying each of the residuals (39) by the coefficient of mi in the ex- pression for that residual, adding the results and equating the sum to zero. The remainder of the I normal equations required are determined with respect to m2, ms, •••, m^ in the same manner. The foregoing processes may be summed up in the following rule : To adjust a set of observation equations of the first degree, write the expression for the residual corre- sponding to each observation equation, multiply it by the coefficient of the first unknown, in that expression, add the products and equate their sum to zero. The result is the normal equation pertaining to the said first unknoivn. Do likewise for each of the other unknowns. Then solve 74 THEORY OF ERRORS AND LEAST SQUARES the I normal equations thus formed for the desired most probable values, mi, m2, "',mi. Let the student prove that taking the arithmetical mean of a number of direct observations upon a single quantity is merely a special application of this rule. 35. Illustrations from Physics. — It will be of material assistance to the student to have presented at this point a number of actual examples illustrating the application of least square adjustment in various departments of exact science. These examples are not " made up " for the purpose ; they are drawn from actual experimental notes on research or field work. 1. Bridge Wire. — It was desired to measure the total resistance of a Wheatstone bridge wire and at the same time to calibrate it, by comparison with a standardized bridge of another type. The unknown (and unessential) resistance of the connections had also to be reckoned with and eliminated. The wire was 100 cm. long, and the meas- urement was conducted by observing the resistance of the first 10 cm., then of the first 20 cm., etc., and finally of the whole wire, the connections entering each time as a con- stant term in the observed resistance. The results follow. No. Cm. Resist., No. Cm. Resist., Measured Ohms Measured Ohms 10 0.116 60 0.595 20 0.205 70 0.675 30 0.295 80 0.760 40 0.388 90 0.850 50 0.503 100 0.926 ADJUSTMENT OF OBSERVATIONS 75 Let X be the total resistance of the bridge wire, and c that of the connections. These are the two unknowns, the first of which is to be obtained with all possible precision, the second to be eliminated, as a mere correction. Mathe- matically they are equally important. The observation equations are ^ ^ , r\ tin ^ 0.1 ar + c = 0.116, 0.2x-{-c = 0.205, O.Sx + c = 0.295, 0.4 X + c = 0.388, 0.5 a: + c = 0.503, 0.6X + C = 0.595, 0.7X + C = 0.675, 0.8 X + c = 0.760, 0.9X + C = 0.850, 1.0 a: + c = 0.926. In practice we need not take the trouble to change symbols in distinguishing between the true and most prob- able values of the unknown (" q " and " m "). If x and c now represent the most probable values sought, the first residual is pi =0.116 — (0.1 a: + c), etc. Let the student apply the rule developed in the preceding article to obtain the two normal equations, which he will find to be 3.85 a: + 5.5 c = 3.686, 5.5 a; + 10 c = 5.313, the solution of which gives X = 0.926 ohms, c = 0.022 ohms. 76 THEORY OF ERRORS AND LEAST SQUARES Let the student select any two of the observation equa- tions and solve them for x and c, comparing the results with these most probable ones. The accompanying figure —•90 —eo — 70 — «0 -50 10 ^ h -s ^ Y K Y ^ l^ ^ ^ —JO ^ ^ ^ —.10 ^ ^ ^ ^ h cms. " lO 1 20 1 30 1 40 1 50 1 60 1 TO 1 80 1 r 100 1 . ..._ J Fig. 6 shows the plotted observations, together with the straight line I 0.926^^ + 0.022 100 R> upon which they all should lie were there no errors in the measurements nor irregularities in the wire itself. The departures of the plotted points from this most probable line represent the residuals of the ten observations. 2. Balance Constants. — The general theory of the equal- arm balance is somewhat complicated, but in the equation used to express the sensibility in terms of the load, the various instrumental constants may all be involved in two quantities a and 6, the equation being a -{-bw = -' s ADJUSTMENT OF OBSERVATIONS 77 Here iv is the load on either pan (grams) and s the gram sensibiHty, or one thousand times the deflection produced by a milligram weight laid on one pan. The constants a and b are to be estimated from the following observations. w s w s Grams Scale Div. Grams Scale Div. 2212 50 2389 10 2265 75 2449 20 2320 100 2563 30 2343 125 2590 40 2316 The observation equations are then a + 6 = 1 H- 2212, a+ 10 6 = 1 -- 2265, a + 20 6 = 1 -^ 2320, a + 30 6 = 1 - 2343, a + 40 6 = 1 ^ 2316, a + 50 6 = 1 ^ 2389, a + 75 6 = 1 - 2449, a + 100 6 = 1 -v- 2563, a + 125 6 = 1 - 2590. The adjustment of these by the foregoing method gives as the most probable values sought, a = -\- 0.0004466, h = - 0.000000518. Let the student perform this reduction. 78 THEORY OF ERRORS AND LEAST SQUARES 36. Illustrations from Chemistry. 1. Volumetric Solutions. — It is desired to test certain acid and alkaline solutions to be used in volumetric chemical analysis, in order to ascertain their exact strengths. Two common reagents, in the form of tenth-normal solu- tions, may be tested first, then others may be compared to these. If the two reagents chosen be hydrochloric acid and potassium hydroxide, the following procedure may be employed. A quantity of each solution is placed in an accurately graduated burette, the two burettes being supported side by side. A small amount (say about 0.2 g.) of finely pulverized pure calcium carbonate (chalk, CaCOs) is carefully weighed, placed in a white porcelain dish and treated with an excess (say about 50 cc.) of the HCl solu- tion from the burette, the amount being accurately ob- served. The chalk dissolves and neutralizes part of the acid, the CO2 gas escaping. The porcelain dish is now set under the KOH burette, and just enough of the alka- line solution allowed to flow into it to render it exactly neutral, this point being determined by a drop or two of methyl orange or other sensitive indicator previously added to the mixture in the dish. The amount of KOH solution thus used is also carefully noted. Part of the acid is neutralized by the CaCOa and the remainder by the KOH. The chemical equations representing the two reactions are as follows: 72.36 99.32 (I) 2 HCl + CaCOa = CaCl2 + CO2 + H2O, ADJUSTMENT OF OBSERVATIONS 79 36.18 55.70 (II) HCl + KOH = KCl + H2O. The small numbers above the symbols are obtained from the molecular weights, and represent the relative weights of the substances engaging in the reaction. Unknown. Let gi = wt. HCl in 1 cc. HCl sol. ^2 = wt. KOH in 1 cc. KOH sol. a — total volume HCl sol. used. a = vol. HCl sol. neutralized by CaCOs. a — a = vol. HCl sol. neutralized by KOH. h = vol. KOH sol. used in neutraUzation. c = wt. CaCOa powder used. Then aqi = wt. HCl neutralized by CaCOa. (a — a)qi = wt. HCl neutralized by KOH. bq2 = wt. KOH used. From (I) aqi'.c -= 72.36 : 99.32 = 0.73, or aqi = 0.73 c. From (II) (a - a)qi : 6^2 = 36.18 : 55.70 = 0.65, or aqi — aqi= 0.65 bq2. From these two equations a is eliminated by addition, giving finally aqi - 0.65 6^2 = 0.73 c. 80 THEORY OF ERRORS AND LEAST SQUARES This is an observation equation, the quantities a, h, c having been measured, and gi, g2 being the two unknowns ; and a series of such experiments (at least two) will yield the most probable values required. In some of the ex- periments the CaCOs powder may be omitted entirely, giving c = ; but not in all of them. (Why ?) • a 6 c Vol. HCI Vol. KOH Wt. CaCOa Sol. used Sol. used Powder used cc. cc. g- 50.00 10.33 0.1779 50.00 7.88 0.1936 11.23 9.98 none 11.25 10.00 none 11.25 10.00 none 11.34 10.10 none The above data yield the following observation equa- tions : 50 gi - 0.65 X 10.33 g2 = 0.73 X 0.1779, 50 qi - 0.65 X 7.88 ^2 = 0.73 X 0.1936, 11.23 gi - 0.65 X 9.98 g2 = 0, 11.25 gi - 0.65 X 10.00 g2 = 0, • 11.25 gi - 0.65 X 10.00 g2 = 0, 11.34 gi -0.65 X 10.10 g2 = 0. Let the student reduce these to normal equations and solve for the most probable values of gi and g2. 2. Pyknometer Constants. — The expansion of a pyk- nometer (specific gravity bottle), like any solid, is in ap- ADJUSTMENT OF OBSERVATIONS 81 proximate accordance with the linear law V = Vo + Kt, V being the capacity at temperature t, Vq the capacity at zero and K a constant involving the coefficient of expan- sion of the glass. The two constants Vq and K must be experimentally determined from time to time for any pyknometer that is used in accurate measurements of density. This may be done by finding the capacity at several different temperatures over the required range.* The following is a tabulation of eight such determinations, using distilled water and corrected for buoyancy of the air. t V t V 19°.20 19.75 25.61 30.92 25.2628 cc. .2634 .2664 .2681 35°.50 39.30 39.75 ^ 46.45 25.2687 cc. .2691 .2692 .2734 Let the student form the eight observation equations and the two normal equations, and reduce for the most prob- able values of Vq and K. (The approximate answers are, T^o = 25.2509, K = 0.0005244.) 37. Illustrations from Surveying. 1. Locating a Distant Station. — Some of the best writers on surveying strongly recommend the use of rec- * If the range be large, K will vary somewhat. The range may be subdivided, say into lO-degree intervals, and the constants found for each ; or better, a quadratic relation assumed, with three constants. See Art. 45. 82 THEORY OF ERRORS AND LEAST SQUARES tangular coordinates in surveying and mapping ; certainly their use reduces many calculations to a more scientific basis. The problem in hand is as follows : Given, the Fig. 7 coordinates of a number of stations A^ B, C, etc., with reference to an origin 0, and the bearing of an unknown station P from each of these stations; to find the most probable coordinates of P. For instance, the unknown ADJUSTMENT OF OBSERVATIONS 83 station P is a reef near the coast along which the points A, B, etc., are located. The numerical data are as follows, for five stations. Station Coordinates (Ft.) Bearing of P Vector Angle B E. N. A 1785 1501 S. 58° 57' W. PA 3r 3' B 1372 2020 S. 22 5 W. PB 67 55 C 1052 1971 S. 5 29 W. PC 84 31 D 909 1609 S. 4 43 E. PD 94 43 E 620 1533 S. 32 43 E. PE 122 43 The vectorial angles in the last column are the angles made by the vectors PA, PB, etc., with the line drawn eastward through P, calculated from the given bearings. Using coordinates x and y to locate P, and Xa, ya, etc., for A, etc., we can write Xa X = tan dn or X tan 6a — y = Xa tan da — yc that is. X tan 31° S' -y = 1785 tan 31° 3' - 1501, etc., as the observation equations, there being as many of these as there are known stations. These equations, being of the first degree in x and y, may be adjusted in the usual manner. Let the student do this. (The results should be, approximately, x = 930, y = 1000 ; that is, P = 930 E., 1000 N.) 84 THEORY OF ERRORS AND LEAST SQUARES 2. Relative Levels of Stations. — The next illustration is taken from Merriman's Least Squares, and is typical of many kinds of measurements in which the quantity sought is measured by parts or segments. The same method is, for example, applied to a number of angles at one station. Given, a number of determinations on the relative altitudes of several stations, obtained by precise leveling, to find the most probable values of their altitudes above one of them taken as a datum. Following are the results of the levelings. A above 573.08 ft. B above A 2.60 ft. B above 575.27 ft. • C above B 167.33 ft. D above C 3.80 ft. D above B 170.28 ft. I) above E 425.00 ft. E above 319.91 ft. (one way) E above 319.75 ft. (another way) Representing by a, 6, etc., the elevations of the respective stations above as a datum, the following simple observa- tion equations at once result. a = 573.08 h-a = 2.60 h = 575.27 c-h = 167.33 d-c = 3.80 ADJUSTMENT OF OBSERVATIONS 85 d-b = 170.28 d-e = 425.00 e = 319.91 e = 319.75 The student can readily adjust these in the usual manner. It will be interesting in this case to compare the adjusted values of a, h and e with their values as directly measured. 38. Illustrations from Astronomy. 1. Errors of the Transit and Clock. — Astronomical time is ascertained, at any observatory, by observations upon the stars. To this end an instrument not unlike a sur- veyor's transit is used. It is larger, however, and fixed on a solid pier, and is incapable of rotating horizontally, being swung in the vertical plane of the meridian. This instrument is the astronomical transit or the meridian circle. When used for time observations, the telescope is set at the proper angle of altitude for some star to traverse its field as it crosses the meridian. The exact sidereal time of meridian passage, or transit, is known as the right ascension"^ of the star, -and is given in the star catalogues. In order to correct the clock, therefore, it is necessary only to note at what time, by the clock, the star is actually ob- served to cross the meridian. * Right ascension on the celestial sphere, as shown by the star maps, is closely analogous to longitude on the earth, only it is usually expressed in hours, minutes and seconds, reading toward the east. Declination corresponds to terrestrial latitude. 86 THEORY OF ERRORS AND LEAST SQUARES On account of the extreme accuracy demanded in as- tronomical work, this apparently simple procedure re- quires the elimination of certain recognized instrumental errors. These are : (1) the level error, arising from the non-horizontality of the bearings or trunnions on which the telescope turns, so that its revolution does not exactly coincide with the meridian plane ; (2) the azimuth error, or failure of this axis of rotation to coincide with the east and west line, which has a similar effect on the plane of rotation ; (3) the collimation error, due to the fact that the cross-wires in the telescope, which determine its line of sight, are not exactly in the optic axis, being a little to one side of the center of the field. In addition to these, there is the error of the clock, which is the quantity really wanted. The level error is ascertained by a direct applica- tion of the stride level resting on the trunnions and having a very sensitive graduated spirit-bubble. The other errors must be found simultaneously from several observations on different stars, the level error reading being simply a part of the determination. Without entering into the applications of spherical as- tronomy required, it may be simply stated that the ob- servation equations involved are of the first degree. If ^1 = the true clock error (clock minus true time), ^2 = the azimuth error, ^3 = the collimation error, / = the level error, all being expressed in seconds of time, then the form of ADJUSTMENT OF OBSERVATIONS 87 the observation equation is Qi + aq2 + cqz = d — bl, d being the apparent clock error, or the time indicated by the clock at apparent transit minus the true time of transit, or right ascension, of the star. The quantities a, b and c are known as Meyer's coefficients, and are calculated from the following formulas : sin (X— 5) a = cos 5 7 _ cos (X— 5) cos 8 c = sec 8, in which X is the latitude of the observatory and 8 the dec- lination of the star used. Tables of these coefficients are at hand in every observatory. Of course three observations on different stars, at least, are required to determine gi and eliminate ^2 and q^. If more are made, least-square reduction may be applied to their adjustment. Following is a typical set of data of this sort, based on the observed transits of six stars. Iowa City, Iowa, Lat. 41° 40' November 16, 1896 Star Declina- tion Right Ascension Observed Time of Transit I a b c « Draconis . . iSCeti . . . y Cassiopeiae . 0" Ursae Majoris f Cygni . . . a Cephei . . 109° 38' -18 33 60 9 112 27 29 48 62 9 h, m. s. 29 4.28 38 26.50 50 30.72 21 1 21.85 21 8 32.81 21 16 6.35 h. m. s. 29 39.14 39 34.07 52 4.77 21 2 1.41 21 9 52.14 21 17 42.64 a. OAT 0.44 0.38 0.59 0.59 0.59 2.76 0.91 -0.64 2.47 0.24 -0.75 -1.12 0.52 1.91 -0.86 1.13 2.01 -2.97 1.05 2.01 -2.61 1.15 2.14 88 THEORY OF ERRORS AND LEAST SQUARES No allowance has here been made for any error in the clock's rate during the progress of the observations. 2. Parallax and Proper Motion. — Stellar parallax is the apparent change in the position of a star, during the year, caused by the earth's motion in its orbit. In addition to this, there is the actual, or '' proper," motion of the star itself through space. These two are superposed and pro- duce one resultant effect upon the star's apparent posi- tion at any time. Their separation into distinguishable components is the problem here presented. Modern astronomical measurements are conducted very largely by photography. The star in question is photographed on the same plate with others so immensely farther away that they have no perceptible parallax or proper motion, and then the positions of the images are measured at leisure on very accurate measuring machines. Let TT = the parallax in a given direction, ft = the proper motion in that direction, s = the measured displacement of the star in tHat direction, with reference to its apparent position at some previous date T days past. Then the observation equation is shown in practical astronomy to be Pt + Tfi -\- c = s. P is the parallax factor, easily calculated from the direc- tion of the star and the position of the earth in its orbit. c is an unknown constant, depending on the f)eculiarities of the measuring machine, and to be eliminated. The ADJUSTMENT OF OBSERVATIONS 89 three unknowns are, then, tt, ft and c. The coefficients P and T and the quantity s are varied by making observa- tions on many different dates, and from the resulting series of observation equations, the most probable values of the proper motion and the parallax are obtained. The latter gives the most probable distance of the star. The details of the process being somewhat technical, no numeri- cal example is here given. 39. Observation Equations Not of First Degree. — If the observation equations are not of the first degree, re- course may be had to the general method explained in Art. 33, that is, to the application of the principle of least squares through the general equations (37). This would often lead, however, to normal equations that would be exceedingly inconvenient to solve. In many such cases, the difficulty may be at once avoided by a suitable application of logarithms. A standard measurement in the physical laboratory, for example, is the simultaneous determination of the magnetic field of the earth H and the magnetic moment M of the bar magnet used for the purpose. One experiment gives the product, ^^^^^^ ^^j^ and another the quotient -^ = S2, (42) of the unknown quantities. These observation equations may be made linear by using instead of M and //, as 90 THEORY OF ERRORS AND LEAST SQUARES (43) unknowns, their common logarithms : log M -[-\ogH = log si, log M — log // = log 52, the most probable values of log M and log H being then found in the usual manner. Again, the solubility of a chemical salt is given by the theoretical formula* ^, s = s^^^^^\ (44) in terms of the centigrade temperature t. Sq is the solubility of the salt at 0° C. and c is a constant depend- ing on its heat of solution, ^o and c are unknowns, to be determined for each substance by means of several meas- urements on s at different temperatures. For this purpose the observation equation may be written t log So + log e ' c = log s, (45) 273 + f the most probable values of c and log ^o being the values directly found. The following data pertain to the solu- bility of potassium chlorate (KCIO3) in water. t 8 (obs.) 8 (calc.) O'' 0.0247 5 0.0299 .0317 10 .0406 .0402 15 .0512 .0507 20 .0672 .0634 25 .0774 .0787 30 .1027 .0970 35 .1145 .1187 40 .1405 .1444 * See Arrhenius, Electrochemistry. ADJUSTMENT OF OBSERVATIONS 91 The eight observations on t and s furnish eight observation equations of the above form, which when adjusted give as most probable values log ^o = —1.6073, whence ^o = 0.0247 ; and c = 13.82. The solubility formula for this substance may now be written in its original form, or more conveniently retained in the logarithmic form : log s = 6.0014 ^ 1.6073, ^ 273 + ^ from which the values of s given in the third column are calculated. The student will find it instructive to plot the observations on t and s, and also the smooth curve corresponding to the calculated values of s. It would be difficult to imagine a more typical application of least- square adjustment than the one just given. Another method of procedure when the observation equations are not of the first degree, somewhat analogous to Horner's method of approximation for higher algebraic equations, is explained in Note B, Appendix. 40. Observations upon Quantities Subject to Rigorous Conditions. — It often happens that unknown quantities involved in observation equations are further connected by known mathematical conditions, which the final ad- justed values must rigorously satisfy. For example, the most probable values of the angles of a triangle could not be a set of angles whose sum is other than exactly 180° ; the sum of all the percentages in a chemical analysis must be 100; etc. Observations upon such quantities are known as conditioned observations. 92 THEORY OF ERRORS AND LEAST SQUARES Suppose that the results of measurements upon the three angles of a triangle are qi = ^2, qz = sz- (46) These are the observation equations. To this list there must be added a fourth equation, namely : 91 + ^2 + gs = 180°, (47) which is called an equation of condition. It differs from the others in that it is known to be exactly true, while the others are not. The three most probable values, when deduced, must satisfy this equation exactly ; the others must be satisfied as nearly as may be. This equa- tion of condition cannot, therefore, be classed as an ob- servation equation and treated like the others. In general, we may have n observations involving / un- knowns, which are further subject to m rigorous conditions, expressed as equations of condition, m must be less than / ; for if equal to it, the unknowns would be absolutely de- termined by the given conditions, and the measurements would be superfluous ; and if greater, no set of quantities could, in general, be found to satisfy all the conditions. There being fewer conditions than unknowns, there is an unlimited number of sets of values of the unknowns which might satisfy the conditions, and we have to de- termine from the n observations which of these sets is the most probable. ADJUSTMENT OF OBSERVATIONS 93 Though the m conditions do not give the values of the / unknowns, they enable us to express m of the unknowns rigorously in terms of the remaining ones; and if we now substitute these expressions for the m unknowns in the observation equations, the latter may then be adjusted for the most probable values of the / — m quantities re- maining. The most probable values of the m replaced quantities may now also be calculated so that the condi- tions are exactly satisfied. Applying this to the case of the angles of a triangle, subject to one condition (47), one of the angles, say gs, may be expressed by means of it in terms of the other ^^'''' gs = 180° - 9i - ^2. (48) The three observation equations then appear: (49) ^2 = -^2, 180° -q,-q2 = 53. Let the student adjust these and show that the most probable values sought are gi = ^i + i[180°- (51 + ^2 4-^3)], ^2 = ^2 + i[180° - (^1 + ^2 + Sz)], (50) qs=Sz-\-i[lS0''-{si-\-S2 + s,)], the third result following from the other two through sub- stitution in (48) ; which shows that the results sought can be obtained by adding to each measured angle one- third the discrepancy between the sum of the measured 94 THEORY OF ERRORS AND LEAST SQUARES angles and 180°, so as to make the sum correct. This is on the assumption that all three of the measurements are equally trustworthy. (See Chap. VII.) The same proceeding is to be followed in every case where the observation equations represent the separately measured values of the unknowns, while the one equation of condi- tion rigorously gives their sum. Cases like this are of common occurrence. If the sides of the triangle are measured, as well as the angles, there will be six observation equations (at least), and three equations of condition. Of these latter, one will be the same as (47), the other two arising from the requirements of trigonometry as to sides and angles. A case of special importance to the surveyor is the ad- justment of the sides and angles of a polygon of land. In addition to whatever measurements are made upon the lengths and bearings of the sides, there are two rigorous conditions to be fulfilled, namely : that the algebraic sum of the projections of the sides on an east-and-west line is zero, and the algebraic sum of their projections on a north-and- south line is zero. This adjustment will be found explained in detail in the more advanced works on plane surveying. EXERCISES 41. 1. Draw a large triangle on paper with a fine pencil, and measure with a protractor each of the angles. Form the observation equations and the equation of condition, and from them deduce the most probable values of the angles. ADJUSTMENT OF OBSERVATIONS 95 2. Lay off on a straight line four points, A, B, C, D. Measure AB, BC, CD, AC, BD, AD, From these measurements form observation equations and compute the most probable values of AB, BC, CD. These seg- ments may be conveniently lettered x, y, z. 3. The following measurements were made upon a rec- tangular metallic tank to determine its dimensions : Length (inside) 27.31 cm. Width (inside) 16.08 cm. Depth (inside) 9.67 cm. Capacity by standard graduates, 4.3217 liters. Find the most probable dimensions. 4. Draw five lines radiating accurately from a common point 0, the further extremities being A, B, C, D, E. Measure with a protractor, by the differential method, and turning the protractor at each measurement, each of the angles AOB, AOC, AOD, AOE, BOC, BOD, BOB, COD, COE, DOE. Determine the most probable values of the angles AOB, BOC, COD, DOE. 5. The following are the results of an analysis of a cer- tain medicinal compound : Salts of calcium 1.26 per cent. Salts of sodium 2.53 per cent. Salts of iron 0.23 per cent. Salts of manganese 0.14 per cent. Salts of quinine 0.07 per cent. 96 THEORY OF ERRORS AND LEAST SQUARES Salts of strychnine 0.02 per cent. Water 95.67 per cent. Find the most probable values of the several percentages. 6. Six points, supposed to lie on the arc of a circle, have the following measured coordinates : X y X y 3.15 2.67 1.80 2.49 3.72 4.69 1.07 -0.20 - 1.84 5.33 5.98 6.25 Find the most probable coordinates of the center and most probable radius. 7. A steel tape was measured under different conditions of stretch and temperature, as follows : Centigrade Temp. Tension, Lb. Obs. Length, Ft. 0° 100.031 20 10 .064 25 8 .068 18 12 .063 21 15 .069 15 15 .062 Using the approximate formula l = lo + at-\- bf, in which t = temperature and / = tension, adjust for the most probable values of h, a, h. ADJUSTMENT OF OBSERVATIONS 97 8. (Adapted from Wright's Adjustment of Observations.) Let D be the difference in length of two standard meter bars at 62° F. and A the difference in their coefficients of expansion. Then the difference d in length at any temperature t is ^ = D -\- (t - Q2) A. Observations were made as follows : t d 24°.7 0.00791 inch 37.1 811 inch 61.7 833 inch 49.3 820 inch 66.8 847 inch 71.5 849 inch Adjust for the most probable values of D and A. 9. Van der Waal's equation for pressure and volume of a gas at absolute temperature T may be put in the form v'^TR — va -{- pv'^b -\- tib = pv^. The measurements of Amagat on air at moderate pressures and at 16° C. (289°. 1 absolute) were published as follows : p IN CM. Mercury pv 76 1.0000 2000 0.9930 2500 .9919 3000 .9908 3500 .9899 4000 .9896 98 THEORY OF ERRORS AND LEAST SQUARES Form the observation equations by the method of Note B, Appendix, and adjust for a, b, R. 10. The electrical conductivity of selenium is found to vary with the intensity of light falling on it according to the equation ,_ The following data were furnished by Dr. F. C. Brown. Intensity / 3 11 17 25 Adjust for the most probable values of a and b. (Note. — In working the above problem, it will be found necessary, as is sometimes the case, to use caution in dropping decimal places, as the normal equations hap- pen to be quite " sensitive " to slight changes in the coefRcients.) 11. The E.M.F. of a thermo-couple for a given tem- perature difference t between junctions may be represented by the equation e = at-\- btK The following values for a copper-tellurium couple with one junction at 0°, in which e is in volts, were furnished by Mr. W. E. Tisdale. CONDU TIVITY c- C Intensity / Conduc- tivity C 83 33 319 188 44 348 250 50 361 285 100 446 303 ADJUSTMENT OF OBSERVATIONS 99 t 50 e t 0.000243 t 150 e 1 0.000254 82 242 162 262 92 245 180 265 100 248 186 267 113 249 190 266 117 250 195 267 128 252 200 268 140 251 Calculate the most probable values of the constants a and b. Plot the curve. 12. In a sine intensity magnetometer, let the pole strength of the bar magnet be P, the distance between its poles /, and the distance from its center to the needle pivot a. 8 is the needle deflection and H the horizontal inten- sity of the earth's magnetism. The equation connecting these quantities is The following data were obtained at a station where H = 0.1884 (c. g.s.). a (cm). 5 a (cm). d 20 24° 17' 40 1° 49' 25 12 46 45 1 35 30 6 48 50 1 27 35 3 26 Find the most probable values of P and /. Use the method of Note B, Appendix. 100 THEORY OF ERRORS AND LEAST SQUARES 13. The specific volume of a certain liquid was meas- ured at different temperatures by a quick secondary method which was known to have certain small persistent errors. At three of the temperatures, known as " tie points," the specific volume was also measured by a more laborious absolute method, free from the said sources of error. The results follow: Temp., C. Sp. Vol., Sp. Vol., Secondary Absolute 23°.0 0.952750 23.5 2879 24.0 3003 24.5 3177 0.953322 25.0 3339 3488 25.5 3505 26.0 3678 26.5 3840 4059 27.0 4012 27.5 4187 28.0 4366 28.5 4526 29.0 4701 29.5 4872 30.0 5075 31.0 5426 In order to correct all the data in the second column, use was made of the equation Y = AX -\- B, in which X is the specific volume by the secondary method and Y the corresponding corrected value, A and B being assumed constant. By using the values of X and Y at ADJUSTMENT OF OBSERVATIONS , Iftl; the three " tie points," find the most probable values of A and B and correct all the secondary data accordingly. 14. A glass sinker, used for precision measurements of liquid density by the Archimedes buoyancy method, was calibrated for expansion, the data being as follows : Temp. C. Vol. Sinker (cc.) 24°.0 34.03894 24.5 3907 25.0 3984 25.5 4085 26.0 4102 26.5 4134 27.0 4191 27.5 4203 28.0 4231 28.5 4240 29.0 4290 30.0 4393 Find the most probable zero volume and coefficient of ex- pansion of the sinker, assuming a linear relation. 16. Guthe and Worthing's formula for the vapor pres- sure of water at temperature ^ C. is logio y = 7.39992 - (^ + 273)' From the following data, find the most probable values of a and h. 10:^ THmRY OF ERRORS AND LEAST SQUARES t p (mm. of Mercury) 10° 9 20 17 30 32 40 55 50 92 60 149 80 355 00 760 16. The angles and the sides of a triangle ABC were measured, with the following results. Angle A 51° 9' 5 95 4 C 33 51 Side 5(7 1721.3 ft. AC 2207.5 AB 1233.0 Introduce the necessary geometric conditions and adjust for the most probable values of sides and angles. 17. Draw accurately a large quadrilateral ABCD and its two diagonals AC and BD. Measure with a millimeter scale the four sides AB, BC, CD, DA, and with a protrac- tor the angles DAB, DAC, CAB, ABC, ABD, DBC, BCD, BCA, ACD, CDA, CDB, BDA. Introduce the eight necessary geometric conditions and adjust for the most probable values of the sides and angles of the quad- rilateral. ADJUSTMENT OF OBSERVATIONS 103 If the diagonals had also been measured, how many conditions would be introduced ? 18. Adjust the transit observations given in Art. 38. 19. (Adapted from Crandall's Geodesy and Least Squares.) Adjust the following transit observation equa- tions for X, y, z. - 0.07 X + 1.41 y -{-z = - 0.65, + 0.68 X + 1.00 y + 2 = + 0.18, + 0.52 X + 1.02 2/ + 2 = + 0.13, + 2.51 X - 2.67 2/ + 2 = + 3.96, - 0.73 X + 2.13 y + z = - 1.88, + 0.75 X + 1.01 y-\-z=-\- 0.02, + 0.53 X + 1.02 y -\-z = -\- 0.13, + 0.68 X + 1.00 y + z = + 0.44, + 0.81 X + 1.02 2/ + 2 = + 0.29, + 0.09 X + 1.27 y -\-z = - 0.76. 20. In the following observation equations the un- knowns n and K are constants of metallic reflection. 4'n-=iiTs^]-"^*- Assuming that approximate values of n and K are known, transform these into observation equations of the first degree by the method of Note B, Appendix. CHAPTER VI EMPIRICAL FORMULAS 42. Classification of Formulas. — If we examine into the many formulas employed to represent natural or physical laws, it is found that they fall into two fairly distinct classes, which may be called, respectively, rational and empirical formulas. To the former class belong those which have been de- duced through processes of mathematical reasoning from the elementary and established laws of the science to which they pertain; hence the term rational. Such, for example, are the equations for the motion of falling bodies, the expressions for electric or gravitational force at a point, the equations of the balance, the error equations for the astronomical transit, etc. In these there appear certain constants or coefficients, the determination of which is often a matter of great scientific importance. Empirical formulas, on the other hand, are those whose form is inferred wholly from the results of experiment or observation, and which have not been deduced theoreti- cally. Some of the best examples of these are to be found in engineering, such as the formulas for the flow of water in pipes and channels, or for steam pressure as a function of temperature. Empirical formulas also contain con- 104 EMPIRICAL FORMULAS 105 stants, which are determined in exactly the same manner as if the formulas were rational, and whose determination depends upon experiment and measurement. A closer examination into the subject reveals, however, the fact that the boundary between these two classes is by no means a sharp one, for the reason that a very large proportion of the rational formulas purporting to represent natural laws have been deduced upon more or less em- pirical and approximate assumptions, which have been adopted for the sake of simplicity of form, or for want of better information. In fact, it may well be doubted whether there exist any absolutely rational formulas per- taining to material magnitudes. Even Newton's great law of gravitation has its experimental basis; and it is possible that some future investigation in astronomy may demonstrate it to be inaccurate. 43. Uses and Limitations of Empirical Formulas. — Empirical formulas owe their existence to the fact that in many cases no rational. formula can be deduced to repre- sent the law of behavior of a phenomenon, but that, nevertheless, experiment shows some law is being obeyed which appears to be simple in character and is therefore presumably expressible, at least approximately, in mathe- matical symbols. Not being able to trace the mechanism operating between cause and effect, on account of its com- plexity or for other reasons, the experimenter must seek more or less blindly for a functional relation that will satisfactorily connect them. It may happen that the 106 THEORY OF ERRORS AND LEAST SQUARES finding of such a relation as accords perfectly with the observations will throw much light on the nature of the mechanism itself, and lead to a theory relative to it, which can be tested by more intelligently directed later ex- periments. Stefan's fourth-power law of cooling, which, though wholly empirical as far as Stefan was concerned, has led to the important modern theory of radiation, is an excellent example of this sort. But the great majority of empirical formulas are con- fessedly artificial, and reveal nothing of the real nature of the connection between the phenomena involved. Many do not even pretend to consistency in the matter of dimen- sions ; the writer has estimated railroad culvert openings, for example, on the crude working rule that the area of opening, in square feet, should be equal to the square root of the drainage area, in acres — an area equal to a length. Nevertheless these formulas are capable of the utmost practical usefulness; for by means of them, depending upon the principle of continuity, we may accurately inter- polate the values of the unknown function between points actually observed, and even, in a limited way, extrapolate beyond the experimental region into conditions unattain- able in practice. There is still another class of empirical formulas, more or less in the nature of scientific curiosities, which repre- sent, in the experimental region only, a relation between variables that have no conceivable connection with each other. It is thus possible to construct an artificial for- mula which will follow, with fair accuracy, the increase EMPIRICAL FORMULAS 107 in population of the United States, or of a city, with time, or even the fluctuations of the stock market over a given interval of time. Such formulas are, however, of little value, as they are merely a sort of cast of a series of statis- tics which are themselves available ; and since the variable represented may not even be continuous, interpolation and extrapolation with any certainty are impossible. It must also be pointed out that empirical formulas cannot be allowed to enter into theoretical developments on the same basis as rational ones, unless their physical nature is first carefully looked into and the region in which they are assumed to apply is properly circumscribed. Where the true functional relation (supposing one to exist) can be dealt with mathematically with safety, an artificial one closely approximating it may lead, if so used, to altogether erroneous conclusions. 44. Illustrations of Empirical Formulas. 1. Reduction of Pendulum to Zero Arc. — The Kater's reversible pendulum is familiar to nearly every physical laboratory student as a means of obtaining the accelera- tion of a faUing body, g, or the value of " gravity." When so adjusted that the time of swing is the same from both supports, i.e., when the knife edges are at conjugate points, the pendulum swings in a period given by the ideal simple pendulum formula in which / is the distance between the knife edges. The 108 THEORY OF ERRORS AND LEAST SQUARES determination of g is therefore a matter of measuring the period of oscillation and the distance I. This equation affords, however, an excellent example of the class referred to in the last paragraph of Art. 42. For in its deduction it is assumed that the pendulum swings without any kind of friction from a perfectly rigid support, and that the amplitude of vibration is infinitely small, none of which conditions is attainable. The writer has attacked these difficulties in the following manner, with good results. Apparatus is arranged to release the pendulum so as to swing with any desired initial angle of amplitude, and the time accurately observed for each of several small amplitudes. The following results, obtained by one of my students, are typical. is the half amplitude in degrees, T the period in seconds. «^ T <}> T 1° 0.878489 6° 0.878807 2 8543 7 8874 3 8622 8 8938 4 8679 9 8975 5 8740 10 9029 The steady increase of period with amplitude includes all factors : the true, theoretical increase that would exist under ideal conditions, and the influences of air friction, pivot friction and bending of supports. The results are plotted in Fig. 8, which shows an unmistakable cur- EMPIRICAL FORMULAS 109 vature with downward concavity. The relation is there- fore not Hnear, but may be approximately quadratic. The empirical formula T = a + b + €'' (51) is now assumed to represent the variable T as a function of Y ^ r\f Tao ) 1 2 S 4 5 6 7 8 9 lO - 0.000001 ^ (52) as the relation desired. In reahty only a is wanted, for it is the value of T for zero amplitude that we are seeking ; that is, the limit approached by T as approaches zero. This is 0.878400 sec. By this slight extrapolation, there- fore, it is possible to extend the experiments into a region unattainable otherwise. The value of g may now be calculated from this result and the measured distance between knife edges. 2. Solubility Formula. — Previous to the theoretical calculation of a rational formula for solubility in terms of temperature (Art. 39), the relation was represented by an empirical formula of simple power terms : s = a + bt -\- cf + dtK (53) The data given in Art. 39 will suffice for the determination of the constants a, 6, c, d, a result being obtained which will fit the observations nearly, if not quite, as well as the rational expression. This exercise is left to the stu- dent. 3. Gordons Formula for Rectangular Columns. — The ul- timate strength of a rectangular column under compres- sion is found to depend fundamentally upon how slender it is ; specifically, upon the ratio of its length to its shorter transverse dimension. For long, slender columns, the relation is found to be expressed satisfactorily by the following formula, in which U is the ultimate compressive EMPIRICAL FORMULAS 111 strength per square inch of cross section, and R the ratio of length to least width. U = (54) a and h are the empirical constants to be determined. The following data refer to white-oak timber columns or posts, U being expressed in pounds per square inch, and will serve as a further exercise for the student. R u R u 10 845 25 585 12 820 28 540 15 770 30 510 18 715 35 435 20. 675 38 400 22 640 40 375 a and h should be about 925 and 0.00091, respectively. 45. Choice of Mathematical Expression. — The reader will now wish to know by what process the form to be used, as representing the unknown relation between variables, may be arrived at. There is no general rule covering this matter. The empirical form once being settled upon, the calculation of the empirical constants is a direct process; but the selection of a mathematical expression which can be made, by the use of proper con- stants, to fit the facts with sufficient accuracy, is often a problem calling for the exercise of the highest degree of ingenuity, especially where there is more than one in- dependent variable. 112 THEORY OF ERRORS AND LEAST SQUARES The first step will probably always be to plot the results of the observations, or the data to be represented, to a suitable scale on coordinate paper. The result will be some sort of curve, which, if at all regular, will give an idea as to the nature of the variation, and will often sug- gest an equation through its resemblance to some well- known locus, such as the straight line, parabola, etc. A few general forms have been found especially adaptable. The equation y = a -\- bx -\- cx^ -{- dx^ + •••, (55) continued so far as may be necessary, may be used for curves which are not periodic, nor asymptotic, nor very irregular. The number of terms to be used will be limited by the fact that the coefficients of powers that may be omitted turn out to be negligibly small. This form was used in two of the three examples in the preceding article, and might probably have been used with some success in the third. It was remarked in connection with the second example of Art. 36 that the volume coefficient of expansion really varies when carried over a considerable range. This might have been allowed for by adding a term involving the square of t, with a third unknown constant coefficient, to those used. When such a form as (55) is used, it will be well to apply to it the values of x and y belonging to five or six of the observed points that seem to lie most accurately on the curve, as a preliminary calculation, and determine from them approximate values of an equal number of the coefficients without least squares, EMPIRICAL FORMULAS 113 in order to ascertain where the series may safely be stopped. It may be found, in this way, that only two or three terms are necessary, the coefficients beyond these being negli- gible. The following equations are also quite adaptable to many physical phenomena, particularly those involving variables which approach a limit, or in which maxima and minima do not appear. y = a + h\og{x + K). (56) a^ = a + 6 log (2/ + K). (57) ax -\- by -\- c = xy. , (58) log y = a -\- b log x. (59) (56) is asymptotic in the y direction, (57) in the x direc- tion; (58) is an equilateral hyperbola, asymptotic in both directions. The logarithmic formulas may easily be put into exponential form if desired. The constant K may sometimes be theoretically assigned. The equation ax -\- by -\- c = x'^y (60) is more general and includes (58). (54) will also be seen to be a special case of it. The curves represented by (60) may have maxima or minima and points of inflection. n may be given a small integral value, as 1, 2, 3, and a, b, c will be the empirical constants to be determined. The student will do well to plot these equations, using assumed values of the constants. For functions that are apparently periodic, or which 114 THEORY OF ERRORS AND LEAST SQUARES have many " ups and downs " in the course of the varia- tion, there may be used a Hmited number of terms of the trigonometric series y = a + b sinnx + c cos nx -\- d sin 2 nx -\- c cos 2 nx + / sin 3 nx + g cos 3 nx -\- •■' . (61) This is a Fourier's series, and can be made to fit any curve with any desired degree of approximation by carrying it to a sufficient number of terms. The calculation of the constants may become extremely laborious, and Prof. A. A. Michelson devised, some years ago, a mechanism known as the harmonic analyzer, which will give their approximate values. By the aid of this machine it is possible to analyze very complicated phenomena, such as the tides or the variations in terrestrial magnetism, into harmonic components, and often to reveal their component causes. But it is also possible, by this means, to express in an altogether artificial manner such phenomena as are referred to toward the close of Art. 43. To empirical formulas of this class applies, particularly, the caution against treating them on the same basis as rational formulas in mathematical analysis. Very often the problem of selecting the proper form will be facilitated by giving attention to obvious limiting con- ditions, such as the fact that eftect is zero where cause is zero, etc. This amounts to making the selection partly rational, and only emphasizes the statement that there is no sharp distinction between rational and empirical ex- EMPIRICAL FORMULAS 115 pressions. After all has been said, however, the student will still find true the remark, previously made, that this matter calls for skill and ingenuity of a high order. EXERCISES 46. 1. Experiments were made upon the index of re- fraction of a solution of varying concentration and density, sodium light being used. The results follow : Density x Index y Density x Index y 1.200 1.378 1.146 1.365 1.187 1.374 1.132 1.361 1.178 1.371 1.123 1.359 1.167 1.369 1.115 1.356 1.156 1.367 1.098 1.352 Express the variation by a suitable empirical formula, deducing the constants. Would it be safe to infer from this formula the index for pure water? 2. A galvanometer attached to a thermo-electric couple gave the following readings y, for the corresponding differences of temperature x: X y X y 0° 45° 5.50 20 2.50 50 6.15 25 3.10 60 7.60 30 3.70 70 8.65 35 4.30 80 9.90 40 4.90 hrf— 90 10.90 Prepare suitable empirical formula, deducing constants. 116 THEORY OF ERRORS AND LEAST SQUARES 3. The following are observed positions of points on a curve : X y X y 5 15.0 1 0.5 6 23.0 2 2.5 7 31.0 3 6.0 8 40.5 4 10.5 9 51.5 Obtain an equation whose graph will fit these points as nearly as possible, and plot it. 4. The temperature of a heated body, cooling in the air, was taken each minute for ten minutes, the results being here tabulated : Time t .Temp. Time t Temp. 84^9 6 6r.9 1 79.9 7 59.9 2 75.0 8 57.6 3 70.7 9 55.6 4 67.2 10 53.4 5 64.3 The temperature of the air was 20°. Deduce an equation expressing 6 in terms of t. 6. Measurements were made upon the radioactivity of a deposit of pure thorium at intervals after its forma- tion, as follows : EMPIRICAL FORMULAS 117 Time Activity Time Activity 10 min. 100.0 5hr. 107.0 20 104.3 6 101.1 40 110.8 8 89.1 60 115.8 10 78.3 80 118.2 12 68.7 100 119.6 15 56.6 120 119.8 18 46.2 3hr. 117.9 20 40.7 4 113.0 Express the relation as an empirical formula, activity will die out with time.) (Note that 6. The quantity of discharge Q, in cubic feet per minute, of a 10-inch sewer pipe was found to vary with the slope (percentage grade) s as per the following data: s Q s 0.1 % 64 2.0% 146 0.2 75 3.0 170 0.4 88 4.0 190 0.6 95 5.0 208 0.8 108 10.0 279 1.0 116 Work out an empirical formula and plot it. 7. The means of many observations upon a certain variable star of short period gave the following variations of magnitude ; 118 THEORY OF ERRORS AND LEAST SQUARES Time (Days) Mag. Time (Days) Maq. 4.65 8 ' 4.20 1 4.10 9 3.57 2 3.50 10 3.70 3 3.80 11 3.93 4 4.00 12 4.07 5 4.10 13 4.35 6 4.40 14 4.64 7 4.65 15 4.40 Represent this variation by as simple a formula as possible. 8. The atmospheric refraction R for a star above the horizon at various altitudes a is given approximately by the following table, corresponding to temperature 50° F. and normal pressure : a R a R 0'' 34' 50" 10° 5' 16" 2 18 6 20 2 37 4 11 37 40 1 9 6 8 23 60 33 8 6 29 90 Represent these as nearly as possible by means of an empirical formula. 9. Amagat's experiments on air at very high pressures gave the following results : EMPIRICAL FORMULAS 119 Press., Atmos. Vol. Press., Atmos. Vol. 1 750 1000 • 1500 1.000000 0.002200 0.001974 0.001709 2000 2500 3000 0.001566 0.001469 - 0.001401 Represent these by an empirical formula. 10. The current through the field coils of a certain dynamo was varied and the voltage generated by the machine simultaneously measured, as follows: Field Current, Armature Field Current, Armature Amps. Volts Amps. Volts 0.000 0.0 1.416 21.0 0.472 8.5 1.650 23.2 0.709 12.1 1.888 25.5 0.943 15.4 2.125 27.3 1.180 18.3 2.360 29.0 Represent these by an empirical formula. 11. The specific gravity of dilute sulphuric acid at different concentrations is given in the following table : CONC. (Per Cent.) Sp. Gray. CONC. (Per Cent.) Sp. Gray. 5 1.033 30 1.218 10 1.068 35 1.257 15 1.101 40 1.300 20 1.139 45 1.345 25 1.178 50 1.389 Represent these by an empirical formula. 120 THEORY OF ERROUS AND LEAST SQUARES 12. A pyknometer being tested for evaporation was allowed to stand in a desiccator and weighed at intervals, as follows : Sept. 30 3 : 15 p.m. 44.4226 grams 4 : 00 P.M. .4223 grams Oct. 2 11:00 a.m. .3855 grams 3 : 30 P.M. .3821 grams Oct. 3 8 : 00 A.M. .3695 grams 4 : 00 P.M. .3622 grams Find the most probable weight at noon October 1. 13. Simultaneous observations were made upon two connected variables x and y with the following results : X y X y 26.5 • 0.002442 31.5 0.005315 27.0 2571 32.0 5607 27.5 2582 32.5 6039 28.0 2885 33.0 6407 28.5 3165 33.5 6947 29.0 3500 34.0 7238 29.5 3738 34.5 7703 30.0 4311 35.0 8092 30.5 4548 35.5 8438 31.0 4991 36.0 8870 Represent these by an empirical formula. 14. Following are vapor pressures, in mm. of mercury, of methyl alcohol at various temperatures : EMPIRICAL FORMULAS 121 t P t P 0'' 30 35° 204 5 40 40 259 10 54 45 327 15 71 50 . 409 20 94 55 508 25 123 60 624 30 159 65 761 Represent these by an empirical formula. 15. Assuming the form log n = logiV — log a log^ 15 15 in which n is per cent, and s is grade, deduce N and a from the data of Ex. 10, Art. 30. Plot the curve. 16. The following average heights and weights for men 35 to 40 years of age were compiled by the medical director of the Connecticut Mutual Life Insurance Co. Height Weight Height Weight 5 ft. in. 131 5 ft. 8 in. 157 1 131 9 162 2 133 10 167 3 136 11 173 4 140 6 179 5 143 1 185 6 147 2 192 7 152 1 ' 200 Represent these by an empirical formula. 122 THEORY OF ERRORS AND LEAST SQUARES 17. The Society for the Promotion of Engineering Edu- cation reports its growth in membership as follows : Year No. Members Year No. Members 1894 156 1905 400 1895 188 1906 415 1896 203 1907 503 1897 226 1908 675 1898 244 1909 747 1899 251 1910 938 1900 266 1911 1040 1901 261 1912 1166 1902 275 1913 1291 1903 326 1914 1358 1904 379 Try to calculate the most probable membership in 1915 from these data. 18. Try to represent the data plotted in Ex. 8, Art. 30, by means of an empirical formula. 19. The following measurements give the average length of the head in schoolboys at different ages (West, Science, Vol. 21, 1893) : Age Length (mm.) Age Length (mm.) 5 176 14 187 6 177 15 188 7 179 16 191 8 180 17 189 9 181 18 192 10 182 19 192 11 183 20 195 12 183 21 192 13 184 Represent these by an empirical formula. EMPIRICAL FORMULAS 123 20. Records of the magnetic declination (departure of compass from the true north) at 25° N. lat., 110° W. long, over a series of years are as follows (U. S. Mag. Tables for 1905) : 1840 9° 28' E. 1875 10° 24' E. 1845 38 1880 25 1850 49 1885 25 1855 10 00 1890 26 1860 09 1895 30 1865 16 1900 36 1870 21 1905 48 Represent these by an empirical formula. CHAPTER VII WEIGHTED OBSERVATIONS 47. Relative Reliability of Observations. Weights. — We have hitherto regarded each one of a set of several observations as having been made with equal mechanical refinement, care and 'skill, and the results as meriting, therefore, the same degree of confidence. This assumption is often, however, far from the truth. The position of a star, for example, as measured with an engineer's transit, 13 less reliable than it would bs if measured with a large meridian circle ; and the results of a series of difficult observations made by a tired research worker in a cold, drafty laboratory are not worth as much as a similar series made by the same person when rested and under favorable conditions. Again, the mean of a long series of careful observations upon a quantity is certainly of more value than the result of a single measurement upon the same quantity. It is therefore evident that, in practical work, it is necessary to employ some means whereby differences in reliability may be taken into account. This can be done by using a method of adjustment in which the more trustworthy results are allowed to have more influence upon the final most probable values than the less reliable 124 WEIGHTED OBSERVATIONS 125 ones, thus giving each result a degree of prominence pro- portional to its reliability. To accomplish this, it is the practice of observers to assign to different observations, numbers, which are sup- posed to represent their relative degrees of reliability, and which are called weights. Thus an observation to which the weight 3 has been assigned is considered to merit only half as much attention in the adjustment as one with the weight 6 ; etc. In order to have some basis of estimation, we may regard an observation of given reliability as being equivalent to the mean of a certain number of observations considered as having standard or unit weight, and this number is the weight of the observation in question. The assign- ment of the weight 10 to an observation means that in the opinion of the observer the result is as trustworthy as the average of ten observations of unit weight. Any standard of trustworthiness may be taken as a unit, but it should be such as to render the weights of all the ob- servations referred to it simple, whole numbers. It is to be remembered that weights are purely relative quantities. The assignment of weights to the several observations of a set is a task demanding the exercise of skill and careful judgment. If each observation is actually the mean of several elementary observations and all are of the same kind, the matter is comparatively simple, since there is in this case a numerical basis of estimate. Other- wise, and especially when the observations are of different kinds, the assignment is not so easy. The problem pre- 126 THEORY OF ERRORS AND LEAST SQUARES sents many analogies to that of giving numerical grades to pupils. Like other processes of the sort, the weighting of ob- servations cannot be covered by any set of definite rules. It may be suggested that the observer should note and record in detail the peculiar circumstances, if any, attend- ing each observation or set of observations which is to enter into the final adjustment, and allow no source of unusual disturbance to go unnoticed. Often it is well to assign weights at the time of the observation, while all the circumstances are fresh in the mind, but this should not take the place of recording the circumstances. It sometimes happens that some one else examines the original notes and prefers to assign weights for himself. I recall a case of this sort, in which the weighting depended solely upon the records which the observer had kept of the weather conditions prevailing at the time of each experi- ment. This was because wind and fluctuations of tem- perature were causes of marked disturbance in this par- ticular work. 48. Adjustment of Observations of Unequal Weight. — In adjusting a set of observations to which different weights have been assigned, we have but to remember that the weight w signifies that the observation in question is the equivalent in importance of w observations of unit weight. It is therefore necessary only to repeat the corresponding observation equation w times, and then proceed as usual with the reduction to normal equations. That is, if the WEIGHTED OBSERVATIONS 127 first observation has weight 2, the second 5, the third 3, etc., then simply write the first observation equation twice, the second five times, the third three times, etc. The number of observation equations is now Xw, the sum of the weights. A simple illustration of this is the case of n direct ob- servations on a single quantity q. If the results are ^i, ^2, •••, s„ with weights wi, W2, •••, Wn, the most probable value as deduced on the above principle is WiSi + W2S2 + h WnSn i^w = T T T ' or m^= -^ — ^ (62) This is called the weighted mean. If all the weights are equal, it becomes simply the mean. With observation equations of the first degree involving several unknowns, the process can be effected by first multiplying the expression for each residual by the coeffi- cient of the unknown contained therein (as in the rule at the close of Art. 34), then multiplying by the weight of the corresponding observation, adding the results and equating the sum to zero, to form the normal equation. In this way each residual is represented in each normal equation a number of times equal to its weight. The same thing may be attained by first multiplying each of the original observation equations by the square root of its weight and then proceeding with the reduction 128 THEORY OF ERRORS AND LEAST SQUARES as usual. These square roots need only be indicated, by means of radical signs, as they will disappear on re- duction. (Let the student show why the square roots of the weights should be thus used, and not the weights themselves.) In the reduction of observations upon quantities limited by conditions (Art. 40), it is evident that the equations of condition are not to be weighted, but only the observation equations. In the process of adjustment, the weighting should be introduced after the conditions have been involved in the observation equations, but before the re- duction of the latter to normal equations. Some of the following examples will illustrate this. EXERCISES 49. 1. Measurements were made upon the segments of a line AB, formed by points C, D upon it, as follows : Mean of 2 observations on AC = 45.10 ft. Mean of 3 observations on AD = 77.96 ft. Mean of 2 observations on CD = 32.95 ft. Mean of 3 observations on CB = 98.36 ft. Mean of 2 observations on DB = 65.55 ft. Mean of 4 observations on AB = 143.55 ft. Find the most probable values of AC, CD, DB. 2. In one time-observation with a transit instrument, only five of the nineteen lines of the reticle were used, viz., Nos. 2, 5, 10, 15, .18. A second observation employed WEIGHTED OBSERVATIONS 129 Pig. 9 all the lines. What can be said as to the relative weights of the two observations, the method of observing being the same in both cases? (Fig. 9.) 3. In determining the constants of a balance, it was borne in mind that the instrument was to be used re- peatedly for the weighing of an ob- ject varying slightly in weight but always in the neighborhood of 43 to 45 grams. Hence the sensibility was measured twenty-five times with a load of 45 grams, giving a mean of 2.402 scale divisions per milli- gram, and only four times with zero load, giving a mean of 2.767 scale divisions. Determine the most probable values of the balance constants (Art. 35, Ex.2). . 4. Draw a triangle and measure its angles with a pro- tractor, one angle being measured but once, the second three times, the third eight times (or some other set of unequal numbers), all the measurements being made differentially. Introduce the necessary condition, assign the proper weights and deduce the most probable values of the angles. 6. The following pointings were made at three sta- tions in the triangulation of California, using a 50- cm. direction theodolite (U. S. Coast Survey Report, 1904) : 130 THEORY OF ERRORS AND LEAST SQUARES Station Pointing on Circle Reading Wt. San Pedro 1 Wilson Peak 1 San Juan 73° 11' 40".97 6.1 118 57 57 .51 6.1 San Juan 1 San Pedro 16 54 50 .29 7.6 I Wilson Peak 84 26 21 .03 7.6 Wilson Peak 1 San Juan 241 39 01 .29 6.7 1 San Pedro 308 21 21 .51 6.7 Adjust for the most probable angles. 6. The range of magnitude of the variable spectroscopic binary star a Geminorum was measured by a selenium photometer on different nights as follows (Stebbins, Astrophysical Journal) : Range Wt. Range Wt. Range Wt. 0.237 5 0.235 3 0.218 4 .217 4 .197 5 .233 4 .233 5 .217 5 .209 3 .231 5 .210 5 .224 5 .217 5 .222 5 .227 5 .205 5 .213 5 .189 3 .207 5 .223 5 .220 5 .227 5 .250 5 .211 4 .231 5 .219 5 Find the weighted mean of these observations. 7. Following are results from precise leveling in Texas (U. S. Coast Survey Report, 1911). The weights assigned are inversely proportional to the squares of the distances between the stations. WEIGHTED OBSERVATIONS 131 Lavernia above Serita . Thomas above Serita Serita above Stockdale . Serita above Ruckman . Stockdale above Ruckman Stockdale above Karnes Ruckman above Karnes Ruckman above Bryde . Karnes above Bryde . . Ruckman above Choate. Bryde above Choate . Bryde above Pettus . . Choate above Pettus . . Bryde above Barroum . Pettus above Barroum . Pettus above Wiess . . Choate above Wiess . . Meters Weight + 57.47 4.8 + 45.73 1.0 + 10.56 2.9 + 33.14 0.6 + 23.62 1.1 + 30.83 0.6 + 6.42 1.8 -11.83 0.9 - 18.66 4.8 + 20.17 0.9 + 32.65 3.6 + 23.34 4.8 - 9.36 10.9 + 11.51 5.1 - 11.71 7.9 + 26.19 7.5 + 17.40 3.3 Adjust for the most probable elevations above the lowest station in the list. 8. Experiments were made for the purpose of rating a Price current meter, used in measuring the velocity of streams. The data are the velocity V of the current in feet per second and the number R of revolutions per second of the meter (Raymond, Plane Surveying). V R Wt. V R Wt. 3.774 1.886 2 1.036 0.466 4.544 2.295 1 1.105 0.503 4.878 2.464 1 7.142 3.678 1.613 0.774 1 2.740 1.342 1.316 0.618 1 6.896 3.552 Assume a linear relation and deduce the two constants. 132 THEORY OF ERRORS AND LEAST SQUARES 9. Four points, y4, 5, C, D, lie consecutively in a straight line. The following distances are measured with a steel tape. AD . . 2871.2 (Ave. of 2) AB . . 1042.0 AC . . 2433.5 BC 1392.2 BD 1828.6 CD . 437.5 Apply the principle that, in chaining, the weights of similarly measured lines are inversely proportional to the squares of their measured lengths, and adjust the above values accordingly. 10. Zenith telescope observations were made at Roslyn Station, Virginia, upon the latitude of that station with various pairs of stars, as follows (Chauvenet, Practical Astronomy). The weights were assigned from the number of observations involved and the precision with which the declinations of the stars employed had been measured. Observed Lat. Wt. Observed Lat. Wt. 37" 14' 24" .78 0.44 37" 14' 25". 15 0.59 25 .05 .67 25 .22 .67 24 .83 .82 24 .84 .67 26 .20 .59 25 .36 .67 25 .91 .43 26 .02 .62 22 .73 .00 25 .42 .44 25 .93 .70 26 .08 .44 25 .18 .65 25 .72 .67 25 .89 1.09 25 .70 1.33 25 .79 1.33 25 .93 1.20 24 .53 0.29 Find the most probable latitude. WEIGHTED OBSERVATIONS 133 11. (Adapted from Chauvenet, Practical Astronomy.) At a station of the U. S. Coast Survey, angles were read on each of four other stations, A, B, C, D, as follows: Angle Wt. Angle Wt. AOB 65° ir 52 ".5 BOC 66 24 15 .6 3 3 COD 87° 2' 24" .7 DOA 141 21 21 .8 3 1 Adjust for the most probable angles. 12. Spectrographic radial velocity measurements were made upon the Orion nebula, using different spectrum lines on different dates, as follows (Lick Observatory Bulletin No. 19) : Date Line Velocity (Km.) Wt. Dec. 8, 1901 Hy + 17.1 3 16 Hp 16.1 2 17 Hp 17.0 2 18 Hy 14.8 3 Find the most probable radial velocity. 13. A certain critical coefficient of expansion was measured several times with different apparatus. Observed Val. Wt. Observed Val. Wt. 0.0045 39 34 30 3 2 5 4 0.0036 26 27 43 2 2 1 3 Find the most probable value from these data. r34 THEORY OF ERRORS AND LEAST SQUARES ' 14. The following data are right ascension corrections to the Berlin Jahrbuch made by the photographic transit at Georgetown Observatory for the star f Ophiuchi on different dates. Cor. Wt. Cor. Wt. Cor. Wt. Cor. Wt. - 0.03 S. 2 + 0.02 S. 2 -0.01 3 + 0.02 3 - .03 3 .00 2 - .04 2 + .02 3 - .01 1 4- .04 1 + .03 2 .00 2 - .02 - .04 1 - .02 3 - .04 3 - .03 1 - .05 1 - .06 2 - .06 3 Find the weighted mean. 16. (Adapted from Wright's Adjustment of Observa- tions.) The following trigonometric levelings were made between two terminal stations A and B, as follows : Stations Meters Wt. Stations Meters Wt. A above 12 914.96 23 3 above 9 216.46 1 A above 10 1287.75 17 5 above 9 899.87 1 A above 11 1299.27 2 5 above 8 1075.77 1 A above 9 1553.09 5 3 above 8 391.74 1 12 above 10 372.73 5 7 above 8 901.78 1 12 above 11 384.41 2 5 above 7 174.45 7 12 above 9 638.30 3 4 above 3 296.69 60 12 above 8 814.35 1 7 above 3 509.49 4 10 above 11 11.60 3 B above 3 1376.19 14 10 above 9 265.48 6 5 above 4 387.24 20 10 above 8 441.10 2 7 above 4 212.75 7 11 above 9 253.87 1 B above 4 1079.50 30 11 above 8 429.55 10 B above 5 692.35 15 9 above 8 175.37 1 WEIGHTED OBSERVATIONS 135 By precise spirit leveling, A was found to be 39.05 meters above B, which may be taken as correct. Adjust the heights of the other stations above B accordingly. 50. Wild or Doubtful Observations. — It sometimes happens that, in the course of a series of measurements, results occur which are so doubtful that the observer is tempted to reject them altogether. In technical language, their weight is so small as to be seemingly negligible, and it is a question whether their retention may not do more harm than good. The doubt may arise from the existence of unusual or disturbing conditions, known to the observer. On one occasion I was making a quantitative analysis to determine the exact concentration of a solution, and during the proc- ess of drying, accidentally spilled a few drops of hydrant water into the residue. My final result was to be an average from the analyses of several specimens, and the accident would unquestionably vitiate the result of this observation ; but the specimens were obtained with diffi- culty and I could ill afford to spare any of the data. Was the result to be rejected or not ? Again, suspicion may be due to a marked difference between the result in question and all the others of the set. This does not refer to mistakes (Art. 9), which may usually be easily rectified. To the observer's best knowl- edge, the doubtful observation deserves as much weight as the others, having been made with the same care ; but he dislikes to retain it, as it is so far out of agreement. 136 THEORY OF ERRORS AND LEAST SQUARES The former class of doubtful observations should, in the opinion of the writer, be rejected unless some idea of the extent of the disturbance can be obtained and due correction made for it if necessary. What I did in the case cited was to test the hydrant water and ascertain that the amount of solids contained in a few drops would not be sufficient to affect the result at all seriously; but I gave only half as much weight to this observation as to the others. With the latter class the case is more doubtful. Just because a result differs from the others is no proof that it is any farther from the truth, especially when the num- ber of observations is small. In casting out such a result, one may be throwing away his most valuable observation. Certain criteria have been proposed for deciding whether to retain or reject a " wild " observation, based upon the law of error distribution. Probably the best decision will be based upon the observer's judgment, it being borne in mind that results of observations should not be tampered with unthinkingly. Where wide deviations occur, it will be well, if possible, to continue the observations until a sufficient number are accumulated to show the law of distribution with some distinctness and symmetry. 51. The Precision Index h. — It was pointed out in Art. 28 that the quantity h in the error equation has to do with the precision of the observations (Art. 13), and that the greater the value of h, the greater is the precision indi- cated, h may thus be termed the '' precision index " WEIGHTED OBSERVATIONS 137 or " measure of precision." We are here naturally led to inquire what connection exists between the precision index and the weight of an observation. For, if we have two sets of measurements, one of which is more precise than the other, the value of h belonging to the error dis- tribution in one set will be larger than that belonging to the other ; while at the same time the weight of one ob- servation from the first set is greater than that of one from the second set. Let hi and Ci be the constants in the equation of error distribution corresponding to the first set, and let Wi be the weight of an observation from that set, supposing them all to have equal weight; and let fh, C2, W2 be the corresponding quantities relating to the second set. The probability of an error x occurring in the first set is 2/1 = cie-'''\ (63) Let the value of the precision index corresponding to a set in which the observations are of unit weight be h. This may be called a " standard index," though no absolute value can as yet be assigned to it. An observation from the first set is equivalent in worth to Wi observations from the standard set, in each of which the probability of an error x is -h^sii y = ce Therefore the probability yi of the error x occurring in the first set is that of its occurring Wi times-in the standard set, which is y^\ giving y^ = ym^^m^-w,h^x^^ (64) 138 THEORY OF ERRORS AND LEAST SQUARES The error x being supposed the same in (63) and (64), and these equations holding for all values of x, comparison gives at once Likewise hi^ = wih^. h2)- ^^^^ Instead of (88) we now have, by substitution of this new expression for h in (87), a/???. € = 0.6745. p^^^, (94) \ n — 1 the important standard formula for the probable error of an observation of unit weight, as obtained from a series of weighted observations. In this formula, before sum- ming the squares of the residuals, each square is multi- plied by the corresponding weight; or, otherwise, each residual is multiplied by the square root of the corre- sponding weight. (See Art. 52.) The same modification may be made in the Peters' formula (89) to adapt it to weighted observations, giving * = 0.8453^^^L, (95) Vn(n — 1) or if w is large, approximately, , = 0.85?^^, (96) n which corresponds to (90). PRECISION 159 EXERCISES 60. 1. Two specific gravity bottles, one of which, No. 7701 a, was of the ordinary type, and the other. No. 7701 c, of a special improved design, were each filled with water five times at the same temperature, the following being the results of the weighings, which were made on the same balance in the same manner : No. 7701 a No. 7701 c 42.602818 45.345518 42.604108 45.345852 42.603512 45.345597 42.602062 45.346437 42.602947 45.346219 Find the probable error of a single filling and weighing with each of the two bottles, and the relative weights of a single observation in the two cases. 2. Eighteen measures of a horizontal angle were made by means of a large Coast Survey theodolite, as follows, the observations being of equal weight : 13° 31' 17''.6 13° 31' 20''.4 21 .5 20 .9 19 .0 23 .5 21 .5 18 .4 26 .2 14 .2 17 .1 21 .0 22 .1 21 .8 20 .1 22 .4 17 .9 17 .6 160 THEORY OF ERRORS AND LEAST SQUARES Find the probable error of a single observation of this series by means of each of the formulas (88), (89) and (90). Regarding the mean as an observation of weight 18, find the probable error of the mean. 3. Find the probable error of one shot in your own target experiment of Art. 11, Ex. 1. 4. Find the probable error of one observation in the series of measurements which you made upon a line in Art. 11, Ex. 2. Also, find the probable error of the mean. 6. Six separate researches, by different observers, upon the velocity of light gave the following mean results, with their probable errors, in kilometers per second : 298000 ± 1000 298500 ± 1000 299930 ± 100 299990 ± 200 300100 ± 1000 299944 ± 50 Assign the proper relative weights and find the probable error of an observation of unit weight. Also, regarding the weighted mean as an observation of weight Ziv, find its probable error. Explain why the answer to the first part of the problem is not 1000, supposing the first observation to be assigned unit weight. From the answer to the second part, do the less precise observations add to the value of the whole? Give reason for your conclusion. PRECISION 161 6. The constant of a Babinet compensator is determined by measuring the distance between two successive dark bands as seen through the analyzer. Micrometer readings were taken as follows : IsT Band 2d Band IsT Band 2d Band 267 225 267 225 269 224 265 227 268 226 268 223 267 227 267 227 264 226 264 226 266 226 266 225 266 227 264 227 268 225 267 226 268 224 266 224 264 225 267 226 Find the probable error of one measurement of the differ- ence in readings; of the mean. 7. Ten measurements were made upon the magnitude of a certain bright star, with the following results : 0.600 0.470 .460 .483 .477 .475 .500 .490 .467 .475 Find the probable error of one measurement and of the mean. 8. Syntheses of carbonic acid gas made from different kinds of carbon by Dumas and Stas gave the following 162 THEORY OF ERRORS AND LEAST SQUARES results (Freund, Cheinical Composition). The numbers represent the percentage of carbon in the gas. Natural Graphite Artificial Graphite Diamond 27.241 .268 .270 .258 .248 27.237 .253 .281 .307 27.251 .276 .301 .263 .275 Find the probable error of one determination and of the mean. 9. In a series of base line measurements made with both steel and invar tapes, the following probable errors were found (U. S. Coast Survey Report, 1907) : Base Line Steel Invar Point Isabel Willamette Tacoma Stephen Brown Valley .... Royalton ONE PART IN 1 300 000 1 730 000 1 630 000 1 120 000 1 420 000 2 260 000 ONE PART IN 2 310 000 3 340 000 2 980 000 2 040 000 3 110 000 2 460 000 Averaging these, find the relative weights of base line measurements made with these two tapes. 10. Apply Peters' formula (95) to find the probable error of an observation of unit weight for the data of Ex. 14, Art. 49. PRECISION 163 61. Probable Errors of Functions of Observed Quanti- ties. — An important phase of the subject of preci- sion is what may be termed the "propagation of error " and illustrated by an example : The prob- able error of the diameter of a circle, obtained by measurement, is c; what is the probable error E of the area calculated therefrom? Or generally, given the probable error of a measured value of a quantity, to find the corresponding probable error of any function of that quantity. Let the measured quantity be g, and the function e=/(9). Let an observation be made upon q with error x^ and let the corresponding error affecting the function Q, as a result of this, be X. Then if x be small, we have ap- proximately X:x = dq:dq, or X='^fx. dq It may now readily be seen that if e and E are the prob- able errors of the measured q and of Q, respectively, then E= ^€. (97) dq This may, however, be shown as follows. If Xi, a'2, •••, Xn are a series of errors committed in measurements upon q, and Xi, X2, •••, Xn are the resulting errors in Q, then as above. 164 THEORY OF ERRORS AND LEAST SQUARES dq Y dQ X =^X " dq ' 2Z2 = ('^Ysx2. (98) or squaring and adding, dq. Now from (87), substituting the value of h given in (78), since the ar's and the Z's are true errors (not residuals), the probable error of ^ is € =0.6745 \/—, and that of Q, E = 0.6745 y from which ^^2 _ e'^ 0.67452' ^Y^= ^^n /SZ2 0.67452 The substitution of these in (98) with subsequent reduction gives (97). That is, the probable error of a function of a single measured quantity is equal to the derivative of the function times the probable error of the measured quantity. For example, if the measured radius of a circle be ^ = 9,67 ± 0.02 cm., the computed area is Q = Tq^ = PRECISION 165 293.7663 sq. cm., and its probable error is E = ± 2 7rg X 0.02 = ± 1.215 sq. cm. In general, Q is a function of several (/) measured quan- tides: «=/(?!, , such that any set of values of Xi, X2, ...,Xn that will render X = xi-\-X2+ '" -\-Xn = ^ (a) will simultaneously render ^ = <^fe) + <^fe)+ - +c^(a:J =0. (6) Let us add a small finite quantity e to any one of the oj's, say Xr, and subtract it from any other, say Xg, making the new values of these quantities Xr = Xr -\- ^y Xa — Xa — e. This will not alter the condition X = 0, and hence will not alter the condition = 0, since, by the hypothe- sis, these conditions are to be simultaneous. This necessi- tates that ^/ X , ^/ X j_r \ \ \ ^r \ ^' *^^^ lct>(Xr + €) - {Xr)] + [cf>iXs " e) - ix,)] = 0, whatever the values of Xr, Xa and e. Dividing through by e, this may be written (l>(Xr + €) - {Xr) _ {xi)=-—(x2)= ••• =3— 0(a;J. dxi ax2 dXn It follows at once that, since the x's may be varied in any manner among themselves, only so condition (a) holds, — (f){x) is a constant, say K. Therefore, integrating, ax (f){x) = Kx + c. That c = follows from (a) and (6) jointly, since substi- tution in (6) gives ^ = K(xi -\-X2 + "' +Xn) +nc = 0, the first term of which vanishes by (a). Hence, necessarily, (f)(x) = Kx, which is Eq. (22), Art. 27. B. Approximation Method for Observation Equations Not of the First Degree. (Supplementary to Art. 39.) — This method requires that the values of the unknowns be very approximately known beforehand, as by choosing such of the observation equations as, when solved simul- taneously, will yield values for all of them. Attention is then given to the unknown small corrections that must be applied to these approximate values; a procedure some- APPENDIX 179 what resembling Horner's method of approximation for algebraic equations. The approximate values, however obtained, being des- ignated by ai, a2, ••*, a^, and the corrections required by q\y q'2, •", q\, the true values of the unknowns are qi = ai + q\, ' 92 = a2 + g'2, qi = G.i + q'l- . Let the non-linear observation equations to be dealt with betypifiedby ;(,^^ ,^, ...,,,) =, (^) The substitution of the values (c) in ((i) gives /(ai + q\, a2 + q'l, •••, a^ + 4i) = ^, W an observation equation in which the unknowns are the small corrections q\y q'2, -") q\. Expanding the first member of this by the general Taylor's theorem, f{o.\-\-q'h ai-^-q'-i, ", 0.1 + q'l) =/(ai, a2, •••, a^) + 9'i— /(ai, a2, •••, ai)+g'2 — /(ai, a2, •••, a^) + ••• oai oa2 + q'l—fi^h a2, •••, a^) + R, aai R being the remainder of the series, which involves higher powers of the very small corrections q'l, etc., and which may therefore he neglected without serious inaccuracy. 180 APPENDIX Denoting fiai; , (12, ' ",a,)hyF, dF 5ai by a, dF 5a2 hyh, oa.1 we may therefore replace {e) by or ag'i + &g'2+-+V, = /, (/) in which s' denotes s — F. This is an observation equation of the first degree, and may be used as such, in combination with other observa- tions similarly obtained from their respective originals, for finding the most probable values of the corrections. This being done, the most probable values of the unknown quantities themselves are found by adding the most probable corrections to the approximate values a. C. Evaluation of the Integral f e-^'^^'dx. (Supple- mentary to Art. 54.) — Equation (70) is I =JJe-^'^'dx. This may be transformed into APPENDIX 181 The new integral V is independent of h. For, let hx = z, then hdx — dz, and /'= Ce-''dz= Ce-'^'dx. (h) Returning, however, to the original form of J' in (g), multiply it by e~^^dh : I'e-^'dh = r"[g-(i+-^)'^^ . hdh]dx. Jo * r and X being independent of /j, we may integrate both members of this equation with respect to h as a parameter, assigning the limits and oo to this integration also, tnUS I /»G0 /»x r /"x ~| rj e-^dh=j J e-^'^^'^^'-hdh\dx. (i) Now the integral within the brackets is readily de- termined : r^-a+x^)h^ . hdh = ^—^ Pg-(i+x^)A^ . 2(1 -^x^)hdh Jo 2(l+a:2)*^o 1 2(1 +x2) Substituting this in (i) gives J ^00 e~^'dh in the first mem- ber is equal to f. Hence ^'^ = T, and from (g) which is equation (72). T — ^'^ 2r 182 APPENDIX D. Evaluation of the Probability Integral. (Supple- mentary to Art. 55.) — The value of the integral y^^C^'e-^'dx (76) may be found when AX < 1 by developing e~^^ into a series and integrating the terms separately. By Mae- laurin's theorem, /^henc 3e = 1- ^^+21- '3! 4! "*i rhx 1 6 -^dx - = AZ {hxy 1 1 {hxy 1 {hxy 3 '2! 5 3! 7 ' (i) This series converges rapidly for values of AZ less than or equal to unity, and may therefore be employed in the calculation of Y in this case. When hX>ly however, it is divergent. We may write /^-^''^^ =/[- i]- 2 --^'^-1 =/[- il'^p- Integrating successively by parts, i^ = 2x 2^ x"- = • 2x 4.7!^ 4-^ x' = - 2 .T 4 .T^ 8 x^ 8 5 Ce-^ J x' dx^' L 2x 4:x^ 8.1 APPENDIX 183 3 . 3-5 16 a: 3'5' 32 a: 7. 3>5.7>9 1 ... "+ 64 x^^ i ^^^ */0 *^AZ (See Note (7.) The value of the integral in this last ex- pression can be found by applying the limits hX and 00 to the successive terms of (k), giving JhX I 9 h + hx l2hX 4:{hXy 3 1 15 1 8 {hxy 16 {hxy •], (m) which converges rapidly when hX>\. Equation (/) will now give values for the integral appearing in Y for this case. Therefore, for }iX<\, use series (j) ; for hX>\, use series (m) substituted in (/). (76) will then yield the values of the probability integral desired. Let the student verify these calculations for, say, hX = ^ and hX = 2. E. Outline of Another Method for Probable Errors of Adjusted Values. (Supplementary to Art. 62.) — The method referred to at the end of Art. 62 is given here 184 APPENDIX without proof. (See Merriman, Method of Least Squares, Art. 74.) Let the normal equations found from the observation equations be Aiwii + Bim2 4- • • • + Rirrii = Ki, A2mi + B2m2 + • • • + i^2^z = K2, AiMi + Bim2 + • • • + Rirrii = Kj, in which the quantities K take the place of Hiaws), etc., in equations (108). Let the literal form of these quanti- ties K be preserved throughout the solution of the normal equations, which will then yield mi = aiKi-{-l3,K2-h"'+\Kt, TTh = a2Ki + A/^2 + ••• + \Ki, ^^=aiKi + 0,K2 + "-+\Kj (n) the quantities a, /3, •••, X being numerical coefficients of the literal quantities K. It may then be shown that the weight of Wi is — , that of Oil m2 is — , that of ma is — , etc. That is, the weight of any ^2 73 most probable value m^ is the reciprocal of the coefficient of the absolute term Kp of the normal equation cor- responding to r)ip as Kp appears in the solution (n) for rUp. The weights of all the most probable values mi, m2, •••, mi are thus calculated. The probable error of an obser- vation of unit weight is given by (107), or, as remarked in Art. 62, may be already known from experience. The APPENDIX 185 probable error of each m may therefore now be found by dividing this standard probable error by the square root of the weight of m (92), as above determined. F. Collection of Important Definitions, Theorems, Rules and Formulas for Convenient Reference. DEFINITIONS Error. — The result of a measurement minus the true value of the quantity measured. (Art. 7.) Residual. — The result of a measurement minus the most probable value of the quantity, as derived from a series of measurements. (Art. 7.) Most Probable Value. — A calculated value of an un- known quantity, based upon the results of measurements, such that the residuals arising therefrom will be most nearly in accord with the normal error distribution. (Arts. 7, 29.) Adjustment. — The process of obtaining from the results of measurements the most probable values of the unknown quantities sought. (Chap. V.) Observation Equation. — An equation, in general only approximately true, connecting one or more unknown quantities, or functions of them, with the result of a measurement. (Art. 31.) Normal Equation. — An equation, in general one of a set of simultaneous equations, whose solution gives the most probable values of the unknowns involved in the observation equations. (Art. 33.) 186 APPENDIX Equation of Condition. — An equation expressing a theo- retical condition which must be exactly satisfied by the cal- culated most probable values of the unknowns. (Art. 40.) Empirical Formula. — A formula expressing a relation between variables, whose mathematical form is inferred from the results of experience or experiment, and which is not deduced theoretically. (Art. 42.) Weights. — Numbers assigned to observations, or to the adjusted values of unknowns, representing the relative degrees of confidence which the respective observations or values are supposed to merit. (Art. 47.) Weighted Mean. — The most probable value of a single unknown quantity obtained by multiplying each obser- vation upon that quantity by its weight, adding the prod- ucts, and dividing by the sum of the weights. (Art. 48.) Probable Error. — A theoretical quantity e, so related to the precision of a system of observations, that the probability of the error of any observation or adjusted value being numerically less than e is equal to the proba- bility of its being numerically greater. (Art. 58.) RULES AND THEOREMS 1. Principle of Least Squares. — (a) The most prob- able value of a measured quantity that can be deduced from a series of direct observations, made with equal care and skill, is that for which the sum of the squares of the residuals is a minimum. (Art. 29.) (b) The most probable value of an unknown quantity that can be deduced from a set of observations upon one APPENDIX 187 of its functions is that for which the sum of the squares of the residuals is a minimum. (Art. 31.) (c) The most probable values of unknown quantities connected by observation equations are those for which the sum of the squares of the residuals of those equations is a minimum. (Art. 33.) {d) The most probable values of unknown quantities connected by weighted observation equations are those for which the sum of the weighted squares of the residuals is a minimum. (Art. 52.) 2. Rules for Adjusting Observation Equations of the First Degree. — (a) Write the expression for the residual corresponding to each observation equation, multiply it by the coefficient of the first unknown, in that expression, add the products, and equate their sum to zero. The result is the normal equation pertaining to the said first unknown. Do likewise for each of the other unknowns. Then solve the normal equations thus formed for the desired most probable values of the unknowns. (Art. 34.) (6) In the case of weighted observation equations, after multiplying the residual by the coefficient of the unknown, multiply again by the weight of the corresponding obser- vation ; then add and proceed as above stated. (Art. 48.) 3. Weight and Precision Index. — The weights of ob- servations are directly proportional to the squares of their precision indices. (Art. 51.) 4. Weight and Probable Error. — The weights of obser- vations are inversely proportional to the squares of their probable errors. (Art. 59.) 188 APPENDIX FORMULAS 1. The Error Equation. (Art. 54.) y = ^e-''^\ (74) 2. Formulas for the Precision Index. (Arts. 57, 59.) (a) For observations of equal precision, standard for- mula, I (6) For weighted observations, standard formula. ^2^7^)- (^^) (c) Peters' formula, disregarding signs of residuals, observations not weighted, j^^^M^-ll^ (82) >/7r2/3 3. Formulas for the Probable Errors of Observations in Terms of Residuals. (Arts. 58, 59.) (a) Probable error of single observation, no weights, standard formula, , — € = 0.6745. /-^^- (88) \ M— 1 (6) With weights assigned, probable error of single ob- servation of unit weight, standard. = 0.6745./^^^^. (94) \ n — 1 APPENDIX 189 (c) For an observation of unit weight, there being I un- known quantities, standard, e = 0.6746./!^. (107) {d) Peters' formulas corresponding to the above (a), (6) and (c), disregarding signs of residuals, € = 0.8453 ^ € = 0.8453-5dM=. (95) € = 0.8453 Vn(n-l) Mn - 1) (e) Simplified Peters' formulas corresponding to the above (a) and (6), adapted to approximate calculation when n is large, disregarding signs of residuals, € = 0.85^. (90) n , = 0.85?^^. (96) n 4. Formulas for Probable Errors of Functions of Quanti- ties, in Terms of Probable Errors of Quantities Themselves, (Art. 61.) (a) Function Q of a single quantity q, E = ^e. (97) dq 190 APPENDIX (6) Function Q of several quantities, qi, q2, •••, Qi, --^/(S^■^(S)'•■■+■■■H^■•■■■ <-'■ (c) Function Q = i^i^i + ^292 + ••• + KiQi, E = 'JK1W + K2W+ •••+K,'€;2. (103) (c?) Function q = Kqi''q2^ -" q{, £ = ^(^)V+(^J,^+...+(5V. (104) 5. Formulas for Probable Errors of Adjusted Values. (Art. 62.) (a) For the arithmetical mean, standard, 0.6745 J—^^. (105) \n(n — 1) (n-1) (6) For the weighted mean, standard, .„„ = 0.6745 J-^(^. (106) \ (n — l)Si/; (c) Peters' formulas corresponding to the above (a) and (6) (Ex. 10, Art. 64), 0.8453 0.8453 Printed in the United States of America. VB 35940 « UNIVERSITY OF CAUFORNIA UBRARY I ill ! ! itj li!! '- 'i'i' ilii J!|!H|L ijiiliHl!!!! ilii jiiiiiiiii